北京科技大学 工科物理实验 大二下

前言

本文由20级学生整理,包括实验目的和仪器、实验原理、实验步骤三个部分。主要是想节约一下大家手机拍照扫描、语音输入或手打的时间。(可能有些任课老师要求手写,那就爱莫能助了)

【5.4 实验原理部分缺失】

使用方法

点击“ 源码 按钮,然后复制粘贴本文相应内容,再次点击源码查看就行

图片部分建议自己画图传上去,不然太明显了。

注意:有部分实验报告的排版有些难看,根据需求可以改一下。

目录

5.1 受迫振动的研究

5.2 霍尔效应

5.3 PN结特性

5.4 液体变温粘滞系数测量及PID使用

5.6 迈克尔逊干涉仪

5.7 全息照相

5.8 用光栅测量光波波长

5.9 光栅光谱仪的使用

5.10 光电效应

5.11 电子电荷e值的测定

5.12 弗兰克-赫兹实验

7.4 超声波在物质中的传播与超声成像

7.6 低真空的获得、测量与直流溅射法制备金属薄膜

5.17 太阳能电池

补充实验1——金属丝线胀系数的测定

补充实验2——偏振光的观察与研究

补充实验3——拉曼光谱实验


5.1 受迫振动的研究

实验目的和仪器

<p><strong>【实验目的】</strong></p><p>(1)研究波尔共振仪中弹性摆轮受迫振动的幅频特性和相频特性。</p><p>(2)研究不同阻尼力矩对受迫振动的影响,观察共振现象。</p><p>(3)学习用频闪法测定相差的方法。</p><p>(4)学习系统误差的修正方法。</p><p><strong>【实验仪器】</strong></p><p>BG-2型波尔共振仪</p>

实验原理

<p>&nbsp; &nbsp; &nbsp; &nbsp;在受迫振动状态下,系统除了受到驱动力的作用外,同时还受到回复力和阻尼力的作用。所以在稳定振动状态时物体的位移、速度变化与驱动力变化不是同相的,存在相差。当驱动力频率接近于系统的固有频率时,物体振动的振幅将增大。当物体振动振幅达到最大时,称为位移共振。位移共振频率接近振动物体固有频率,但比振动物体固有频率小。阻尼越小,位移共振频率越接近振动物体固有频率。当驱动力频率与振动物体固有频率相同时,受迫振动的速度幅达到最大,产生速度共振,此时物体振动位移比驱动力滞后90&deg;。</p><p>&nbsp; &nbsp; &nbsp; &nbsp;本实验所采用的波尔共振中的摆轮可在弹性力矩作用下自由摆动。若同时加上阻尼力矩和驱动力矩,摆轮可做受迫振动。用其来研究受迫振动特性,可直观地显示机械振动中的一些物理现象。</p><p>&nbsp; &nbsp; &nbsp; &nbsp;当摆轮受到周期性驱动外力矩M=M<sub>0</sub>cosw<span class="mq-math-mode" latex-data="\omega" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1146px;"><var mathquill-command-id="3">&omega;</var></span></span>&nbsp;t的作用,并在有空气阻尼和电磁尼的介质中运动时(阻尼力矩设为-<span class="mq-math-mode" latex-data="\gamma\frac{d\Theta}{dt}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 64.3646px;"><var mathquill-command-id="5">&gamma;</var><span class="mq-fraction mq-non-leaf" mathquill-command-id="7" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="8"><var mathquill-command-id="11">d</var><span mathquill-command-id="12">&Theta;</span></span><span class="mq-denominator" mathquill-block-id="9" style="width: 37.8229px;"><var mathquill-command-id="15">d</var><var mathquill-command-id="23">t</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;,<span class="mq-math-mode" latex-data="\gamma" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 15.7917px;"><var mathquill-command-id="5">&gamma;</var></span></span>&nbsp;为尼力矩系数),其运动方程为J<span class="mq-math-mode" latex-data="\frac{d^2\Theta}{dt^2}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 64.9062px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="24" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="25"><var mathquill-command-id="33">d</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="30" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="31"><span mathquill-command-id="34">2</span></span></span><span mathquill-command-id="35">&Theta;</span></span><span class="mq-denominator" mathquill-block-id="26" style="width: 50.1562px;"><var mathquill-command-id="37">d</var><var mathquill-command-id="43">t</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="40" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="41"><span mathquill-command-id="44">2</span></span></span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;=-k<span class="mq-math-mode" latex-data="\Theta" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1042px;"><span mathquill-command-id="45">&Theta;</span></span></span>&nbsp;-<span class="mq-math-mode" latex-data="\gamma\frac{d\Theta}{dt}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 64.3646px;"><var mathquill-command-id="47">&gamma;</var><span class="mq-fraction mq-non-leaf" mathquill-command-id="49" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="50"><var mathquill-command-id="53">d</var><span mathquill-command-id="54">&Theta;</span></span><span class="mq-denominator" mathquill-block-id="51" style="width: 37.8229px;"><var mathquill-command-id="56">d</var><var mathquill-command-id="57">t</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;+M<sub>0</sub>cos<span class="mq-math-mode" latex-data="\omega" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1146px;"><var mathquill-command-id="58">&omega;</var></span></span>&nbsp;t(5.1-1)式中。J为摆轮的转动惯量:-k<span class="mq-math-mode" latex-data="\Theta" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1042px;"><span mathquill-command-id="60">&Theta;</span></span></span>&nbsp;为弹性力矩;k为弹簧的劲度系数;M<sub>0</sub>为驱动力矩的幅值;<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&omega;</span>为驱动力的角频率。令<span class="mq-math-mode" latex-data="\omega_0^2" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.8229px;"><var mathquill-command-id="70">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="67" style="font-size: 26.91px;"><span class="mq-sup" mathquill-block-id="65"><span mathquill-command-id="72">2</span></span><span class="mq-sub" mathquill-block-id="68"><span mathquill-command-id="73">0</span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 16.56px; text-align: center; white-space: nowrap;">&Theta;</span>=<span class="mq-math-mode" latex-data="\frac{k}{J}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 32.0729px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="74" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="75"><var mathquill-command-id="78">k</var></span><span class="mq-denominator" mathquill-block-id="76" style="width: 17.3229px;"><var mathquill-command-id="79">J</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;,2&beta;=<span class="mq-math-mode" latex-data="\frac{\gamma}{J}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 32.0729px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="74" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="75"><var mathquill-command-id="80">&gamma;</var></span><span class="mq-denominator" mathquill-block-id="76" style="width: 17.3229px;"><var mathquill-command-id="79">J</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;,m=<span class="mq-math-mode" latex-data="\frac{M_0}{J}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 54.875px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="74" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="75"><var mathquill-command-id="93">M</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="90" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="91"><span mathquill-command-id="94">0</span></span><span style="display: inline-block; width: 0px;">​</span></span></span><span class="mq-denominator" mathquill-block-id="76" style="width: 40.125px;"><var mathquill-command-id="79">J</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;则式(5.1-1)变为&nbsp;当<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 16.56px; text-align: center; white-space: nowrap;">mcos</span><span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&omega;t</span>=0时,即在无周期性驱动外力矩作用时,式(5.1-2)即为阻尼振动方程;且当阻尼系数&beta;=0,即无阻尼时,式(5.1-2)变为简谐振动方程,<span class="mq-math-mode" latex-data="\omega_0" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.8229px;"><var mathquill-command-id="121">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="118" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="119"><span mathquill-command-id="123">0</span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;即为振动系统的固有角频率。&nbsp; 方程(5.1-2)的通解为</p><p><span class="mq-math-mode" latex-data="\Theta" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1042px;"><span mathquill-command-id="124">&Theta;</span></span></span>&nbsp;=<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Theta;<sub>1</sub></span>e<sup>-&beta;t</sup>cos(<span class="mq-math-mode" latex-data="\omega" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1146px;"><var mathquill-command-id="126">&omega;</var></span></span>&nbsp;<sub>1</sub>t+<span class="mq-math-mode" latex-data="\alpha" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 19.6979px;"><var mathquill-command-id="128">&alpha;</var></span></span>&nbsp;)+<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Theta;</span><sub>2</sub>cos(<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&omega;</span>t+&phi;) (5.1-3)</p><p>由式(5.1-3)可见,受迫振动可分成两部分:第一部分,<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Theta;<sub>1</sub></span>e<sup>-&beta;t</sup>cos(<span class="mq-math-mode" latex-data="\omega" style="font-size: 18.4px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 17px;"><var mathquill-command-id="126">&omega;</var></span></span>&nbsp;<sub>1</sub>t+<span class="mq-math-mode" latex-data="\alpha" style="font-size: 18.4px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 13.6625px;"><var mathquill-command-id="128">&alpha;</var></span></span>&nbsp;)表示阻尼振动,经过一定时间后振动衰减至可忽略不计。第二部分,因驱动力矩对摆轮做功,向振动系统传送能量,使系统最终达到稳定的动状态。此时振幅不变,其值为</p><p><span class="mq-math-mode" latex-data="\Theta_2" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.8125px;"><span mathquill-command-id="137">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="134" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="135"><span mathquill-command-id="139">2</span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;=<span class="mq-math-mode" latex-data="\frac{m}{\sqrt{\left(\omega_0^2-\omega^2\right)^2+4\beta^2\omega^2}}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 292.417px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="140" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="141"><var mathquill-command-id="144">m</var></span><span class="mq-denominator" mathquill-block-id="142" style="width: 277.667px;"><span class="mq-non-leaf" mathquill-command-id="145"><span class="mq-scaled mq-sqrt-prefix" style="transform: scale(1, 1.95556);">&radic;</span><span class="mq-non-leaf mq-sqrt-stem" mathquill-block-id="146"><span class="mq-non-leaf" mathquill-command-id="207"><span class="mq-scaled mq-paren" style="transform: scale(1.16667, 2.2);">(</span><span class="mq-non-leaf" mathquill-block-id="189"><var mathquill-command-id="188">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="194" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="196"><span mathquill-command-id="195">2</span></span><span class="mq-sub" mathquill-block-id="192"><span mathquill-command-id="191">0</span></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="198">&minus;</span><var mathquill-command-id="200">&omega;</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="202" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="204"><span mathquill-command-id="203">2</span></span></span></span><span class="mq-scaled mq-paren" style="transform: scale(1.16667, 2.2);">)</span></span><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="184" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="185"><span mathquill-command-id="209">2</span></span></span><span class="mq-binary-operator" mathquill-command-id="210">+</span><span mathquill-command-id="211">4</span><var mathquill-command-id="217">&beta;</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="214" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="215"><span mathquill-command-id="219">2</span></span></span><var mathquill-command-id="232">&omega;</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="229" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="230"><span mathquill-command-id="234">2</span></span></span></span></span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;(5.1-4)</p><p>它与驱动力矩之间的相差<span class="mq-math-mode" latex-data="\varphi" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 20.5417px;"><var mathquill-command-id="235">&phi;</var></span></span>&nbsp;为<span class="mq-math-mode" latex-data="\varphi" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 14.1875px;"><var mathquill-command-id="235">&phi;</var></span></span>&nbsp;=arctan<span class="mq-math-mode" latex-data="\frac{2\beta\omega}{\omega_0^2-\omega^2}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 114.448px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="237" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="238"><span mathquill-command-id="241">2</span><var mathquill-command-id="242">&beta;</var><var mathquill-command-id="244">&omega;</var></span><span class="mq-denominator" mathquill-block-id="239" style="width: 99.6979px;"><var mathquill-command-id="254">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="251" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="249"><span mathquill-command-id="256">2</span></span><span class="mq-sub" mathquill-block-id="252"><span mathquill-command-id="257">0</span></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="258">&minus;</span><var mathquill-command-id="264">&omega;</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="261" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="262"><span mathquill-command-id="266">2</span></span></span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;=arctan<span class="mq-math-mode" latex-data="\frac{\beta T_0^2T}{\pi\left(T^2-T_0^2\right)}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 148.667px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="237" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="238"><var mathquill-command-id="242">&beta;</var><var mathquill-command-id="275">T</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="272" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="270"><span mathquill-command-id="276">2</span></span><span class="mq-sub" mathquill-block-id="273"><span mathquill-command-id="277">0</span></span><span style="display: inline-block; width: 0px;">​</span></span><var mathquill-command-id="278">T</var></span><span class="mq-denominator" mathquill-block-id="239" style="width: 133.917px;"><span class="mq-nonSymbola" mathquill-command-id="281">&pi;</span><span class="mq-non-leaf" mathquill-command-id="283"><span class="mq-scaled mq-paren" style="transform: scale(1.16667, 2.2);">(</span><span class="mq-non-leaf" mathquill-block-id="284"><var mathquill-command-id="292">T</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="289" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="290"><span mathquill-command-id="293">2</span></span></span><span class="mq-binary-operator" mathquill-command-id="294">&minus;</span><var mathquill-command-id="286">T</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="251" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="249"><span mathquill-command-id="256">2</span></span><span class="mq-sub" mathquill-block-id="252"><span mathquill-command-id="257">0</span></span><span style="display: inline-block; width: 0px;">​</span></span></span><span class="mq-scaled mq-paren" style="transform: scale(1.16667, 2.2);">)</span></span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;(5.1-5)由式(5.1-4)和式(5.1-5)可看出,稳定振动状态的振幅<span class="mq-textarea" style="font-size: 18.4px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: text; box-sizing: border-box; position: relative; font-family: Symbola, &quot;Times New Roman&quot;, serif;"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: text; box-sizing: border-box; position: absolute; clip: rect(1em, 1em, 1em, 1em); transform: scale(0); resize: none; width: 1px; height: 1px;" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.425px; font-size: 18.4px; line-height: inherit; margin: 0px; border-color: black; user-select: none; font-family: Symbola, &quot;Times New Roman&quot;, serif;"><span mathquill-command-id="137" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="134" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em;"><span class="mq-sub" mathquill-block-id="135" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><span mathquill-command-id="139" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">2</span></span><span style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; width: 0px;">​</span></span></span>与相差的数值<span class="mq-math-mode" latex-data="\varphi" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 14.1875px;"><var mathquill-command-id="235">&phi;</var></span></span>&nbsp;取决于驱动力矩的幅值M<sub>0</sub>、驱动力的频率<span class="mq-math-mode" latex-data="\omega" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1146px;"><var mathquill-command-id="295">&omega;</var></span></span>&nbsp;、系统的固有频率<span class="mq-math-mode" latex-data="\omega_0" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.8229px;"><var mathquill-command-id="302">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="299" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="300"><span mathquill-command-id="304">0</span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;和阻尼系数&beta;四个因素,而与振动的起始状态无关。</p><p>受迫振动的振幅与驱动力频率有关,由极大值条件<span class="mq-math-mode" latex-data="\frac{\delta\Theta}{\delta\omega}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 51.6354px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="305" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="306"><var mathquill-command-id="311">&delta;</var><span mathquill-command-id="313">&Theta;</span></span><span class="mq-denominator" mathquill-block-id="307" style="width: 36.8854px;"><var mathquill-command-id="317">&delta;</var><var mathquill-command-id="315">&omega;</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;=0可知,当驱动力角频率为</p><p><span class="mq-math-mode" latex-data="\omega_r" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 35.5938px;"><var mathquill-command-id="324">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="321" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="322"><var mathquill-command-id="326">r</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;=<span class="mq-math-mode" latex-data="\sqrt{\omega_0^2-2\beta^2}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 156.208px;"><span class="mq-non-leaf" mathquill-command-id="327"><span class="mq-scaled mq-sqrt-prefix" style="transform: scale(1, 1.75);">&radic;</span><span class="mq-non-leaf mq-sqrt-stem" mathquill-block-id="328"><var mathquill-command-id="338">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="335" style="font-size: 26.91px;"><span class="mq-sup" mathquill-block-id="333"><span mathquill-command-id="340">2</span></span><span class="mq-sub" mathquill-block-id="336"><span mathquill-command-id="341">0</span></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="342">&minus;</span><span mathquill-command-id="343">2</span><var mathquill-command-id="349">&beta;</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="346" style="font-size: 26.91px;"><span class="mq-sup" mathquill-block-id="347"><span mathquill-command-id="351">2</span></span></span></span></span></span></span>&nbsp;(5.1-6)</p><p><span class="mq-math-mode" latex-data="\Theta_r" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 35.5833px;"><span mathquill-command-id="357">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="354" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="355"><var mathquill-command-id="359">r</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;=<span class="mq-math-mode" latex-data="\frac{m}{2\beta\sqrt{\omega_0^2-\beta^2}}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 170.479px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="360" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="361"><var mathquill-command-id="364">m</var></span><span class="mq-denominator" mathquill-block-id="362" style="width: 155.729px;"><span mathquill-command-id="365">2</span><var mathquill-command-id="366">&beta;</var><span class="mq-non-leaf" mathquill-command-id="368"><span class="mq-scaled mq-sqrt-prefix" style="transform: scale(1, 1.73333);">&radic;</span><span class="mq-non-leaf mq-sqrt-stem" mathquill-block-id="369"><var mathquill-command-id="379">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="376" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="374"><span mathquill-command-id="381">2</span></span><span class="mq-sub" mathquill-block-id="377"><span mathquill-command-id="382">0</span></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="383">&minus;</span><var mathquill-command-id="390">&beta;</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="386" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="387"><span mathquill-command-id="389">2</span></span></span></span></span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;(5.1-7)</p><p>&nbsp; &nbsp; &nbsp; &nbsp;此时,系统产生共振。阻尼系数&beta;越小,共振时驱动力角频率越接近系统固有角频率,振幅就越大。</p><p><img alt="" height="512" src="/files/testpaper/106/2022/03-31/201701d10575308923.jpg" width="400" /></p>

实验步骤

<p><strong>1. 定性观察摆轮的自由振动和阻尼振动</strong></p><p>选中进入&ldquo;自由振荡&rdquo;(或&ldquo;阻尼振荡&rdquo;,再选中&ldquo;阻尼选择&rdquo;),在&ldquo;测量&rdquo;置于默认的&ldquo;关&rdquo;状态下,可以直接从屏幕上读出每次振动的振幅和周期,注意观察振幅变化的特点.&nbsp;数据的自动记录与查询功能:当测量界面中的&ldquo;测量&rdquo;处于被选择时,可以更改(通过上下按键,以下同)它的状态,将其置于&ldquo;开&rdquo;状态,则控制箱可以在摆轮摆动时自动记录屏上所显示的数据。当&ldquo;测量&rdquo;回到&ldquo;关&rdquo;状态时,可以选中&ldquo;回查&rdquo;,进入查询界面,通过上下按键查看所有记录的数据,按&ldquo;确认&rdquo;按钮可以退出回查状态。</p><p><strong>&nbsp;2. 测定受迫动的幅频特性和相频特性曲线(至少完成一种阻尼下的测量)</strong></p><p>&nbsp; &nbsp; &nbsp; &nbsp;选中&ldquo;阻尼振荡&rdquo;,将&ldquo;阻尼选择&rdquo;选中于适当的档位(如&ldquo;阻尼2&rdquo;),再由测量界面返回到&ldquo;实验步骤&rdquo;界面后,才能选中&ldquo;强迫振荡&rdquo;进行测量。注意在实验过程中&ldquo;阻尼选择&rdquo;不能任意改变,或将整机电源切断,否则由于电磁铁剩磁现象将引起&beta;值变化,只有在某一阻尼系数&beta;的所有实验数据测量完毕,需要改变&beta;值时,才可改变&ldquo;阻尼选择&rdquo;。进人&ldquo;强迫振荡&rdquo;的测量界面(默认选择&ldquo;电机&rdquo;)后,更改&ldquo;电机&rdquo;状态令其置于&ldquo;开&rdquo;,则启动电动机,当保持&ldquo;周期&rdquo;为&ldquo;1&rdquo;时,屏上可以直接显示摆轮和电动机的周期、振幅值。改变电动机的转速.即改变驱动力矩的频率w。设定某一电动机转速,当受迫振动稳定后,才可以开始准备测量(此时摆轮和电动机的周期必须趋向一致)。选择&ldquo;周期&rdquo;,把&ldquo;周期&rdquo;更改为&ldquo;10&rdquo;,再选择&ldquo;测量&rdquo;,更改其状态为&ldquo;开&rdquo;,控制箱开始自动记录数据。一次测量完成,测量&rdquo;状态显示&ldquo;关&rdquo;,读出摆轮的振幅值0。记录驱动力矩10次振动周期10T、按住&ldquo;闪光灯&rdquo;按钮,利用闪光灯测定受迫振动位移与驱动力间的相差。将所测电动机转速刻度值、驱动力周期10T、振幅、相差等数据记录。</p><p>&nbsp; &nbsp; &nbsp; &nbsp;每次改变强迫力周期钮的刻度(即改变电动机转速)进行测量前,均需返回&ldquo;周期&rdquo;为1的测量界面,等待系统稳定,之后再进行相应的测量.&nbsp;强迫振动测量完毕、选中&ldquo;返回&rdquo;,回到&ldquo;实验步骤&rdquo;选择界面</p><p><strong>&nbsp;3. 测定尼系数&beta;</strong></p><p>&nbsp; &nbsp; &nbsp; &nbsp;在&ldquo;实验步骤&rdquo;界面选中&ldquo;阻尼振荡&rdquo;,将&ldquo;阻尼选择&rdquo;置于与&ldquo;强迫振荡&rdquo;测量时相同的档位,确认进入阻尼测量界面。将有机玻璃盘上零度标志放在0位置,用手将摆轮转动140&deg;~150&deg;左右。松手后将&ldquo;测量&rdquo;状态更改为&ldquo;开&rdquo;。控制箱开始自动连续记录摆轮做阻尼振动10次的振幅数值<span class="mq-math-mode" latex-data="\Theta" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1042px;"><span mathquill-command-id="3">&Theta;</span></span></span><sub>&nbsp;0</sub>、<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Theta;<sub>1</sub>、&Theta;<sub>2</sub>、&hellip;&hellip;、&Theta;<sub>n</sub></span>。及周期。在&ldquo;测量&rdquo;回到&ldquo;关&rdquo;时,可以利用回查功能查询记录的数据。测量数据记录,利用公式ln<span class="mq-math-mode" latex-data="\frac{\Theta_0e^{-\beta t}}{\Theta_0e^{-\beta\left(t+nT\right)}}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 176.042px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="6"><span mathquill-command-id="14">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="11" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="12"><span mathquill-command-id="16">0</span></span><span style="display: inline-block; width: 0px;">​</span></span><var mathquill-command-id="23">e</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="20" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="21"><span mathquill-command-id="24">&minus;</span><var mathquill-command-id="25">&beta;</var><var mathquill-command-id="27">t</var></span></span></span><span class="mq-denominator" mathquill-block-id="7" style="width: 161.292px;"><span mathquill-command-id="28">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="30" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="32"><span mathquill-command-id="31">0</span></span><span style="display: inline-block; width: 0px;">​</span></span><var mathquill-command-id="34">e</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="36" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="38"><span mathquill-command-id="37">&minus;</span><var mathquill-command-id="39">&beta;</var><span class="mq-non-leaf" mathquill-command-id="44"><span class="mq-scaled mq-paren" style="transform: scale(0.997531, 1.18519);">(</span><span class="mq-non-leaf" mathquill-block-id="45"><var mathquill-command-id="47">t</var><span class="mq-binary-operator" mathquill-command-id="48">+</span><var mathquill-command-id="49">n</var><var mathquill-command-id="50">T</var></span><span class="mq-scaled mq-paren" style="transform: scale(0.997531, 1.18519);">)</span></span></span></span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;=n&beta;t=ln<span class="mq-math-mode" latex-data="\frac{\Theta_0}{\Theta_n}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 51.4479px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="6"><span mathquill-command-id="14">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="11" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="12"><span mathquill-command-id="16">0</span></span><span style="display: inline-block; width: 0px;">​</span></span></span><span class="mq-denominator" mathquill-block-id="7" style="width: 36.6979px;"><span mathquill-command-id="28">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="30" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="32"><var mathquill-command-id="51">n</var></span><span style="display: inline-block; width: 0px;">​</span></span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;(5.1-8)求出&beta;式中,n为阻尼振动的周期次数;<span class="mq-math-mode" latex-data="\Theta_n" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.5625px;"><span mathquill-command-id="58">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="56"><var mathquill-command-id="60">n</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;为第n次振动的振幅,T为阻尼振动周期的平均值(可以测出10个摆轮振动周期值,取其平均值)。重复2、3次。</p><p><strong>&nbsp;4. 测定振幅<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Theta;</span>与固有频率<span class="mq-math-mode" latex-data="\omega_0" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.8229px;"><var mathquill-command-id="61">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="56"><span mathquill-command-id="63">0</span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;的对应关系</strong></p><p>将有机玻璃盘上零度标志线保持在&ldquo;0&rdquo;处,选中&ldquo;自由振荡&rdquo;测量。用手将摆轮提动到较大偏转处(约140&deg;~150&deg;)后放手,可以直接从屏上读出每次振幅值及其相应的摆动周期。若振幅变小时,周期不变,则可不必记录,也即只记录周期值变化时对应的幅值。重复几次即可作出<span class="mq-math-mode" latex-data="\Theta_n" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.5625px;"><span mathquill-command-id="64">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="56"><var mathquill-command-id="66">n</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;与T<sub>0n</sub>的对应关系。请注意、记录振幅幅值范围应该涵盖上表所测量的振幅值。</p><p>&nbsp; &nbsp; &nbsp; &nbsp;如果选择控制箱自动记录数据的测量功能(此时只记录摆轮周期值变化时对应的振幅值),则可以利用回查功能对振幅和周期值进行查询</p><p>在&ldquo;实验步骤&rdquo;界面,持续按住复位钮几秒钟,仪器自动复位,实验数据全部清除,关闭电源,实验结束。</p><p><strong>5.数据记录和处理</strong></p><p>1)绘制幅频特性和相频特性测量曲线:分别求出阻尼系数&beta;和各个振幅所对应的固有振动周期T<sub>0</sub>(频率<var mathquill-command-id="61" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif; white-space: nowrap;">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; font-family: Symbola, &quot;Times New Roman&quot;, serif; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="56" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><span mathquill-command-id="63" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">0</span></span><span style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; width: 0px;">​</span></span>)。作出幅频特性和相频特性曲线(至少作出一种阻尼下的特性曲线)。</p><p>2)计算阻尼系数&beta;</p><p>&nbsp; &nbsp; &nbsp; &nbsp;①利用作图法或直线拟合最小二乘法求出&beta;值:根据式(5.1-8)、利用表中的数据,采用作图法或直线拟合最小二乘法得到线性关系In(<span class="mq-math-mode" latex-data="\Theta_n" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span></span><span class="mq-math-mode" latex-data="\Theta_0" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.8125px;"><span mathquill-command-id="64">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="56"><span mathquill-command-id="67">0</span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;/<span class="mq-math-mode" latex-data="\Theta_n" style="font-size: 29.9px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 38.5625px;"><span mathquill-command-id="64">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="56"><var mathquill-command-id="66">n</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;)~n,由该直线的斜率求出&beta;值。</p><p>&nbsp; &nbsp; &nbsp; &nbsp;②利用共振曲线确定&beta;值:当受迫振动达到稳定状态时,阻尼振动部分可以认为已衰减至零,因而振动位移只需要考虑强迫振动部分、即式(5.1-3)的第二项。阻尼系数较小(满足&beta;<sup>2</sup>&le;<span class="mq-math-mode" latex-data="\omega_0" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 38.8229px;"><var mathquill-command-id="68">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="56"><span mathquill-command-id="67">0</span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;<sup>2</sup>)时,在共振位置附近(<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&omega;</span>=<var mathquill-command-id="68" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif; white-space: nowrap;">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; font-family: Symbola, &quot;Times New Roman&quot;, serif; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="56" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><span mathquill-command-id="67" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">0</span></span></span>),由于<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&omega;+</span><var mathquill-command-id="68" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif; white-space: nowrap;">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; font-family: Symbola, &quot;Times New Roman&quot;, serif; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="56" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><span mathquill-command-id="67" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">0</span></span></span>=2<var mathquill-command-id="68" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif; white-space: nowrap;">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="55" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; font-family: Symbola, &quot;Times New Roman&quot;, serif; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="56" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><span mathquill-command-id="67" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">0</span></span></span>,从式(5.1-4)和式(5.1-7)可得出<span class="mq-math-mode" latex-data="\left(\frac{\Theta}{\Theta_r}\right)^2" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 94.125px;"><span class="mq-non-leaf" mathquill-command-id="105"><span class="mq-scaled mq-paren" style="transform: scale(1.2, 2.82);">(</span><span class="mq-non-leaf" mathquill-block-id="103"><span class="mq-fraction mq-non-leaf" mathquill-command-id="94" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="96"><span mathquill-command-id="95">&Theta;</span></span><span class="mq-denominator" mathquill-block-id="98" style="width: 33.7917px;"><span mathquill-command-id="97">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="99" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="101"><var mathquill-command-id="100">r</var></span><span style="display: inline-block; width: 0px;">​</span></span></span><span style="display: inline-block; width: 0px;">​</span></span></span><span class="mq-scaled mq-paren" style="transform: scale(1.2, 2.82);">)</span></span><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="90" style="font-size: 26.91px;"><span class="mq-sup" mathquill-block-id="91"><span mathquill-command-id="107">2</span></span></span></span></span>&nbsp;=<span class="mq-math-mode" latex-data="\frac{4\beta^2\omega_0^2}{4\omega_0^2\left(\omega-\omega_0\right)^2+4\beta^2\omega_0^2}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 290.615px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="109" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="110"><span mathquill-command-id="113">4</span><var mathquill-command-id="119">&beta;</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="116" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="117"><span mathquill-command-id="121">2</span></span></span><var mathquill-command-id="130">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="127" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="125"><span mathquill-command-id="132">2</span></span><span class="mq-sub" mathquill-block-id="128"><span mathquill-command-id="133">0</span></span><span style="display: inline-block; width: 0px;">​</span></span></span><span class="mq-denominator" mathquill-block-id="111" style="width: 275.865px;"><span mathquill-command-id="134">4</span><var mathquill-command-id="151">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="148" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="146"><span mathquill-command-id="153">2</span></span><span class="mq-sub" mathquill-block-id="149"><span mathquill-command-id="154">0</span></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-non-leaf" mathquill-command-id="186"><span class="mq-scaled mq-paren" style="transform: scale(1.08889, 1.73333);">(</span><span class="mq-non-leaf" mathquill-block-id="176"><var mathquill-command-id="175">&omega;</var><span class="mq-binary-operator" mathquill-command-id="177">&minus;</span><var mathquill-command-id="179">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="181" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="183"><span mathquill-command-id="182">0</span></span><span style="display: inline-block; width: 0px;">​</span></span></span><span class="mq-scaled mq-paren" style="transform: scale(1.08889, 1.73333);">)</span></span><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="171" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="172"><span mathquill-command-id="188">2</span></span></span><span class="mq-binary-operator" mathquill-command-id="189">+</span><span mathquill-command-id="190">4</span><var mathquill-command-id="192">&beta;</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="194" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="196"><span mathquill-command-id="195">2</span></span></span><var mathquill-command-id="198">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="204" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="206"><span mathquill-command-id="205">2</span></span><span class="mq-sub" mathquill-block-id="202"><span mathquill-command-id="201">0</span></span><span style="display: inline-block; width: 0px;">​</span></span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;=<span class="mq-math-mode" latex-data="\frac{\beta^2}{\left(\omega-\omega_0\right)^2+\beta^2}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 200.531px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="109" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="110"><var mathquill-command-id="119">&beta;</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="116" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="117"><span mathquill-command-id="121">2</span></span></span></span><span class="mq-denominator" mathquill-block-id="111" style="width: 185.781px;"><span class="mq-non-leaf" mathquill-command-id="186"><span class="mq-scaled mq-paren" style="transform: scale(1.08889, 1.73333);">(</span><span class="mq-non-leaf" mathquill-block-id="176"><var mathquill-command-id="175">&omega;</var><span class="mq-binary-operator" mathquill-command-id="177">&minus;</span><var mathquill-command-id="179">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="181" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="183"><span mathquill-command-id="182">0</span></span><span style="display: inline-block; width: 0px;">​</span></span></span><span class="mq-scaled mq-paren" style="transform: scale(1.08889, 1.73333);">)</span></span><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="171" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="172"><span mathquill-command-id="188">2</span></span></span><span class="mq-binary-operator" mathquill-command-id="189">+</span><var mathquill-command-id="192">&beta;</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="194" style="font-size: 24.219px;"><span class="mq-sup" mathquill-block-id="196"><span mathquill-command-id="195">2</span></span></span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;</p><p>当<span class="mq-math-mode" latex-data="\Theta" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1042px;"><span mathquill-command-id="208">&Theta;</span></span></span>&nbsp;=<span class="mq-math-mode" latex-data="\frac{1}{\sqrt{2}}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 68.3438px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="210" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="211"><span mathquill-command-id="214">1</span></span><span class="mq-denominator" mathquill-block-id="212" style="width: 53.5938px;"><span class="mq-non-leaf" mathquill-command-id="215"><span class="mq-scaled mq-sqrt-prefix" style="transform: scale(1, 0.955556);">&radic;</span><span class="mq-non-leaf mq-sqrt-stem" mathquill-block-id="216"><span mathquill-command-id="218">2</span></span></span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;<span class="mq-math-mode" latex-data="\Theta_r" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 35.5833px;"><span mathquill-command-id="224">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="221" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="222"><var mathquill-command-id="226">r</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;日,即<span class="mq-math-mode" latex-data="\left(\frac{\Theta}{\Theta_r}\right)^2" style="font-size: 18.4px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 59.45px;"><span class="mq-non-leaf" mathquill-command-id="105"><span class="mq-scaled mq-paren" style="transform: scale(1.2, 2.82);">(</span><span class="mq-non-leaf" mathquill-block-id="103"><span class="mq-fraction mq-non-leaf" mathquill-command-id="94" style="font-size: 16.56px;"><span class="mq-numerator" mathquill-block-id="96"><span mathquill-command-id="95">&Theta;</span></span><span class="mq-denominator" mathquill-block-id="98" style="width: 20.7875px;"><span mathquill-command-id="97">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="99" style="font-size: 14.904px;"><span class="mq-sub" mathquill-block-id="101"><var mathquill-command-id="100">r</var></span><span style="display: inline-block; width: 0px;">​</span></span></span><span style="display: inline-block; width: 0px;">​</span></span></span><span class="mq-scaled mq-paren" style="transform: scale(1.2, 2.82);">)</span></span><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="90" style="font-size: 16.56px;"><span class="mq-sup" mathquill-block-id="91"><span mathquill-command-id="107">2</span></span></span></span></span>&nbsp;=<span class="mq-math-mode" latex-data="\frac{1}{2}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 33.8333px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="227" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="228"><span mathquill-command-id="231">1</span></span><span class="mq-denominator" mathquill-block-id="229" style="width: 19.0833px;"><span mathquill-command-id="232">2</span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;时,由上式可得&nbsp;<var mathquill-command-id="175" style="font-size: 16.56px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif; text-align: center; white-space: nowrap;">&omega;</var><span class="mq-binary-operator" mathquill-command-id="177" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px 0.2em; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; font-family: Symbola, &quot;Times New Roman&quot;, serif; text-align: center; white-space: nowrap;">&minus;</span><var mathquill-command-id="179" style="font-size: 16.56px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif; text-align: center; white-space: nowrap;">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="181" style="font-size: 14.904px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; font-family: Symbola, &quot;Times New Roman&quot;, serif; text-align: center; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="183" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><span mathquill-command-id="182" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">0</span></span><span style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; width: 0px;">​</span></span>=+<span style="font-size: 13.3333px;">(-)&beta;</span>。&nbsp; &nbsp;作幅频特性曲线<span class="mq-non-leaf" mathquill-command-id="105" style="font-size: 18.4px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; font-family: Symbola, &quot;Times New Roman&quot;, serif; white-space: nowrap;"><span class="mq-scaled mq-paren" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px 0.1em; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: top; transform-origin: center 0.06em; transform: scale(1.2, 2.82);">(</span><span class="mq-non-leaf" mathquill-block-id="103" style="font-size: inherit; line-height: inherit; margin: 0.1em 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="94" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px 0.2em; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; text-align: center; vertical-align: -0.4em;"><span class="mq-numerator" mathquill-block-id="96" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px 0.1em; border-color: black; user-select: none; box-sizing: border-box; display: block;"><span mathquill-command-id="95" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">&Theta;</span></span><span class="mq-denominator" mathquill-block-id="98" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0.1em; border-top: 1px solid; border-right-color: black; border-bottom-color: black; border-left-color: black; user-select: none; box-sizing: border-box; display: block; float: right; width: 20.7875px;"><span mathquill-command-id="97" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="99" style="font-size: 14.904px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em;"><span class="mq-sub" mathquill-block-id="101" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><var mathquill-command-id="100" style="font-size: inherit; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif;">r</var></span><span style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; width: 0px;">​</span></span></span><span style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; width: 0px;">​</span></span></span><span class="mq-scaled mq-paren" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px 0.1em; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: top; transform-origin: center 0.06em; transform: scale(1.2, 2.82);">)</span></span><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="90" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: 0.5em; font-family: Symbola, &quot;Times New Roman&quot;, serif; white-space: nowrap;"><span class="mq-sup" mathquill-block-id="91" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: text-bottom;"><span mathquill-command-id="107" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">2</span></span></span>-<span class="mq-math-mode" latex-data="\omega" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 25.1146px;"><var mathquill-command-id="233">&omega;</var></span></span>&nbsp;,对应于图中<span class="mq-non-leaf" mathquill-command-id="105" style="font-size: 18.4px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; font-family: Symbola, &quot;Times New Roman&quot;, serif; white-space: nowrap;"><span class="mq-scaled mq-paren" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px 0.1em; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: top; transform-origin: center 0.06em; transform: scale(1.2, 2.82);">(</span><span class="mq-non-leaf" mathquill-block-id="103" style="font-size: inherit; line-height: inherit; margin: 0.1em 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="94" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px 0.2em; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; text-align: center; vertical-align: -0.4em;"><span class="mq-numerator" mathquill-block-id="96" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px 0.1em; border-color: black; user-select: none; box-sizing: border-box; display: block;"><span mathquill-command-id="95" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">&Theta;</span></span><span class="mq-denominator" mathquill-block-id="98" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0.1em; border-top: 1px solid; border-right-color: black; border-bottom-color: black; border-left-color: black; user-select: none; box-sizing: border-box; display: block; float: right; width: 20.7875px;"><span mathquill-command-id="97" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="99" style="font-size: 14.904px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em;"><span class="mq-sub" mathquill-block-id="101" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><var mathquill-command-id="100" style="font-size: inherit; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif;">r</var></span><span style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; width: 0px;">​</span></span></span><span style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; width: 0px;">​</span></span></span><span class="mq-scaled mq-paren" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px 0.1em; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: top; transform-origin: center 0.06em; transform: scale(1.2, 2.82);">)</span></span><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="90" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: 0.5em; font-family: Symbola, &quot;Times New Roman&quot;, serif; white-space: nowrap;"><span class="mq-sup" mathquill-block-id="91" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: text-bottom;"><span mathquill-command-id="107" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">2</span></span></span>=<span class="mq-math-mode" latex-data="\frac{1}{2}" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 22.3375px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="227" style="font-size: 16.56px;"><span class="mq-numerator" mathquill-block-id="228"><span mathquill-command-id="231">1</span></span><span class="mq-denominator" mathquill-block-id="229" style="width: 11.7375px;"><span mathquill-command-id="232">2</span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;,<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&omega;</span>有两个值<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&omega;<sub>1</sub>、&omega;</span><sub>2</sub>。通常把共振幅频特性曲线上相对强度1/2处曲线的宽度定义为共振峰的宽度或共振带宽。由此求出&beta;=<span class="mq-math-mode" latex-data="\frac{\omega_2-\omega_1}{2}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 114.448px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="235" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="236"><var mathquill-command-id="247">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="244" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="245"><span mathquill-command-id="249">2</span></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="241">&minus;</span><var mathquill-command-id="255">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="252" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="253"><span mathquill-command-id="257">1</span></span><span style="display: inline-block; width: 0px;">​</span></span></span><span class="mq-denominator" mathquill-block-id="237" style="width: 99.6979px;"><span mathquill-command-id="258">2</span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;</p><p>&nbsp;&nbsp;3)求解品质因数Q</p><p>&nbsp; &nbsp; &nbsp; &nbsp;①对于无驱动欠阻尼振动系统,振动的振幅满足关系式<span class="mq-math-mode" latex-data="\Theta=\Theta_1" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 95.1042px;"><span mathquill-command-id="259">&Theta;</span><span class="mq-binary-operator" mathquill-command-id="261">=</span><span mathquill-command-id="267">&Theta;</span><span class="mq-supsub mq-non-leaf" mathquill-command-id="264" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="265"><span mathquill-command-id="269">1</span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;e<sup>-&beta;t</sup>,则其能量(正比于振幅平方)满足关系式E=E<sub>1</sub>e<sup>-2&beta;t</sup>。振动系统的能量减小到初始能量的1/e所经历的时间为<span class="mq-math-mode" latex-data="\tau" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 14.7083px;"><var mathquill-command-id="270">&tau;</var></span></span>&nbsp;=<span class="mq-math-mode" latex-data="\frac{1}{2\beta}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 47.2396px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="272" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="273"><span mathquill-command-id="276">1</span></span><span class="mq-denominator" mathquill-block-id="274" style="width: 32.4896px;"><span mathquill-command-id="277">2</span><var mathquill-command-id="278">&beta;</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;称为时间常量,或鸣响时间。技术上定义鸣响时间内振动次数的2&pi;倍为阻尼振动的品质因数Q,Q=2<span class="mq-math-mode" latex-data="\pi" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 19.1042px;"><span class="mq-nonSymbola" mathquill-command-id="280">&pi;</span></span></span>&nbsp;<span class="mq-math-mode" latex-data="\frac{\tau}{T}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 35.0938px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="282" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="283"><var mathquill-command-id="286">&tau;</var></span><span class="mq-denominator" mathquill-block-id="284" style="width: 20.3438px;"><var mathquill-command-id="288">T</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;=<span class="mq-math-mode" latex-data="\omega\tau" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 35.8229px;"><var mathquill-command-id="289">&omega;</var><var mathquill-command-id="291">&tau;</var></span></span>&nbsp;。阻尼不严重时,可用振动系统的固有周期或频率计算。②对于共振系统,可由幅频特性共振曲线确定Q值,即Q=<span class="mq-math-mode" latex-data="\frac{\omega_0}{共振带宽}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 127.792px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="293" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="294"><var mathquill-command-id="302">&omega;</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="299" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="300"><span mathquill-command-id="304">0</span></span><span style="display: inline-block; width: 0px;">​</span></span></span><span class="mq-denominator" mathquill-block-id="295" style="width: 113.042px;"><span mathquill-command-id="305">共</span><span mathquill-command-id="307">振</span><span mathquill-command-id="309">带</span><span mathquill-command-id="311">宽</span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;</p><p><img alt="" height="411" src="/files/testpaper/106/2022/03-31/201717dbef00623693.jpg" width="400" /></p>

5.2 霍尔效应

实验目的和仪器

<p><strong>【实验目的】</strong></p><p>(1)学习霍尔效应的原理和霍尔效应实验中的副效应及其消除方法。</p><p>(2)了解半导体的导电特性,学习确定半导体试样的导电类型、载流子浓度以及迁移率的实验方法。</p><p><strong>【实验方法】</strong></p><p>QS-H型霍尔效应实验组合仪、半导体(硅)样品、导线等。</p>

实验原理

<p><strong>1、霍尔效应</strong></p><p>&nbsp; 将一个通电导体置于磁场中,如图所示,磁场B垂直于电流<span class="mq-math-mode" latex-data="I_s" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 24.4375px;"><var mathquill-command-id="9">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="10">s</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;的方向,则在导体中垂直于B与<span class="mq-math-mode" latex-data="I_s" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 24.4375px;"><var mathquill-command-id="9">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="10">s</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;的方向上出现一个电势差<span class="mq-math-mode" latex-data="U_H" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 45.0312px;"><var mathquill-command-id="12">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="14">H</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;,这个现象称为霍尔效应。以N型半导体为例,将一个N型半导体薄片放置在垂直于它的磁场中,如图。磁场B的方向沿Z轴方向,在X的反方向通以电流<span class="mq-math-mode" latex-data="I_s" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 24.4375px;"><var mathquill-command-id="15">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="16">s</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;,此时N型半导体内的多数载流子电子以一定的速度v沿X方向运动,垂直的磁场会对运动的电子产生一个洛伦兹力的作用,即<span class="mq-math-mode" latex-data="F_B=q\left(v\times B\right)" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 192.333px;"><strong><var mathquill-command-id="17">F</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="18">B</var></span><span style="display: inline-block; width: 0px;">​</span></span></strong><span class="mq-binary-operator" mathquill-command-id="19">=</span><var mathquill-command-id="20">q</var><span class="mq-non-leaf" mathquill-command-id="21"><span class="mq-scaled mq-paren" style="transform: scale(1, 1.2);">(</span><span class="mq-non-leaf" mathquill-block-id="22"><strong><var mathquill-command-id="24">v</var></strong><span class="mq-binary-operator" mathquill-command-id="29">&times;</span><strong><var mathquill-command-id="31">B</var></strong></span><span class="mq-scaled mq-paren" style="transform: scale(1, 1.2);">)</span></span></span></span>&nbsp;。由于洛伦兹力的作用,电荷出现横向偏转并在样品边界处累积,产生一个横向的电场E。当载流子所受电场力与洛伦兹力相等时,样品两侧电荷积累将达到动态平衡。</p><p>即<span class="mq-math-mode" latex-data="qE=q\left(v\times B\right)" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 189.344px;"><var mathquill-command-id="32">q</var><strong><var mathquill-command-id="34">E</var></strong><span class="mq-binary-operator" mathquill-command-id="19">=</span><var mathquill-command-id="20">q</var><span class="mq-non-leaf" mathquill-command-id="21"><span class="mq-scaled mq-paren" style="transform: scale(1, 1.2);">(</span><span class="mq-non-leaf" mathquill-block-id="22"><strong><var mathquill-command-id="24">v</var></strong><span class="mq-binary-operator" mathquill-command-id="29">&times;</span><strong><var mathquill-command-id="31">B</var></strong></span><span class="mq-scaled mq-paren" style="transform: scale(1, 1.2);">)</span></span></span></span>&nbsp;。如果该N型半导体薄片的载流子浓度为n,样品薄片宽度为b,厚度为d,则有<span class="mq-math-mode" latex-data="I=nqvbd" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 122.26px;"><var mathquill-command-id="35">I</var><span class="mq-binary-operator" mathquill-command-id="36">=</span><var mathquill-command-id="37">n</var><var mathquill-command-id="38">q</var><var mathquill-command-id="39">v</var><var mathquill-command-id="40">b</var><var mathquill-command-id="41">d</var></span></span>&nbsp;。</p><p>综合前面式子得<span class="mq-math-mode" latex-data="U_H=R_H\frac{IB}{d}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 159.458px;"><var mathquill-command-id="48">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="44" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="45"><var mathquill-command-id="49">H</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="50">=</span><var mathquill-command-id="58">R</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="54" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="55"><var mathquill-command-id="57">H</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-fraction mq-non-leaf" mathquill-command-id="59" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="60"><var mathquill-command-id="63">I</var><var mathquill-command-id="64">B</var></span><span class="mq-denominator" mathquill-block-id="61" style="width: 30.7812px;"><var mathquill-command-id="65">d</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;。<span class="mq-math-mode" latex-data="R_H=\frac{1}{nq}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 119.938px;"><var mathquill-command-id="58">R</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="54" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="55"><var mathquill-command-id="57">H</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="70">=</span><span class="mq-fraction mq-non-leaf" mathquill-command-id="59" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="60"><span mathquill-command-id="69">1</span></span><span class="mq-denominator" mathquill-block-id="61" style="width: 32.2917px;"><var mathquill-command-id="66">n</var><var mathquill-command-id="67">q</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;为霍尔系数。对于有厚度得霍尔器件,采用霍尔灵敏度<span class="mq-math-mode" latex-data="K_H=\frac{1}{nqd}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 135.073px;"><var mathquill-command-id="71">K</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="54" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="55"><var mathquill-command-id="57">H</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="70">=</span><span class="mq-fraction mq-non-leaf" mathquill-command-id="59" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="60"><span mathquill-command-id="69">1</span></span><span class="mq-denominator" mathquill-block-id="61" style="width: 45.75px;"><var mathquill-command-id="66">n</var><var mathquill-command-id="67">q</var><var mathquill-command-id="73">d</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;表示霍尔器件灵敏度。</p><p><strong>2、半导体材料的参数测量</strong></p><p>(1)载流子浓度 如果待测半导体材料只有一种载流子导电且所有载流子具有相同漂移速度,则载流子浓度n为<span class="mq-math-mode" latex-data="n=\frac{1}{R_Hq}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 117.656px;"><var mathquill-command-id="75">n</var><span class="mq-binary-operator" mathquill-command-id="70">=</span><span class="mq-fraction mq-non-leaf" mathquill-command-id="59" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="60"><span mathquill-command-id="69">1</span></span><span class="mq-denominator" mathquill-block-id="61" style="width: 52.7604px;"><var mathquill-command-id="81">R</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="78" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="79"><var mathquill-command-id="82">H</var></span><span style="display: inline-block; width: 0px;">​</span></span><var mathquill-command-id="83">q</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;。</p><p>(2)载流子迁移率<span class="mq-math-mode" latex-data="\mu" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 19.0104px;"><var mathquill-command-id="84">&mu;</var></span></span>&nbsp;&nbsp;电导率<span class="mq-math-mode" latex-data="\sigma" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span></span><span class="mq-math-mode" latex-data="\sigma=\frac{I_SL}{U_{AC}bd}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 147.365px;"><var mathquill-command-id="86">&sigma;</var><span class="mq-binary-operator" mathquill-command-id="88">=</span><span class="mq-fraction mq-non-leaf" mathquill-command-id="89" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="90"><var mathquill-command-id="109">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="105" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="106"><var mathquill-command-id="108">S</var></span><span style="display: inline-block; width: 0px;">​</span></span><var mathquill-command-id="115">L</var></span><span class="mq-denominator" mathquill-block-id="91" style="width: 82.6771px;"><var mathquill-command-id="112">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="95" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="96"><var mathquill-command-id="110">A</var><var mathquill-command-id="111">C</var></span><span style="display: inline-block; width: 0px;">​</span></span><var mathquill-command-id="113">b</var><var mathquill-command-id="114">d</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;,电导率与迁移率的关系为<span class="mq-math-mode" latex-data="\mu=\left|R_H\right|\sigma" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 130.594px;"><var mathquill-command-id="116">&mu;</var><span class="mq-binary-operator" mathquill-command-id="88">=</span><span class="mq-non-leaf" mathquill-command-id="125"><span class="mq-scaled mq-paren" style="transform: scale(1.09, 1.74);">|</span><span class="mq-non-leaf" mathquill-block-id="126"><var mathquill-command-id="130">R</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="122" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="123"><var mathquill-command-id="128">H</var></span><span style="display: inline-block; width: 0px;">​</span></span></span><span class="mq-scaled mq-paren" style="transform: scale(1.09, 1.74);">|</span></span><var mathquill-command-id="118">&sigma;</var></span></span>&nbsp;。</p><p><strong>3、霍尔效应中的副效应</strong></p><p>&nbsp; 上述推导过程是理想情况,实际上会有各种副效应产生的附加电压叠加在霍尔电压上,因此必须考虑副效应的影响。</p><p>(1)由于不等位电势引起的副效应 由于制造工艺的影响,测量霍尔电压的电极很难做到在一个理想的等势面上。当电流通过该样品时,即使不加磁场,也会产生附加电压。</p><p>(2)热磁副效应1)埃廷斯豪效应:当电流通过霍尔样品时,速度大的载流子受洛伦兹力作用偏向一侧,使得半导体两侧温度不同,从而产生温差电动势,并叠加在霍尔电压测量上。2)能斯特效应:当霍尔样品的电流两端电极与基地接触电阻不同,将产生不同的焦耳热并造成温度梯度。沿着该温度梯度扩散的载流子受到磁场的作用而偏转,产生电位差,并叠加到霍尔测量上。3)里吉-勒迪克效应:由于能斯特效应沿着温度梯度扩散的载流子受磁场作用而发生偏转,而速度不同的载流子使得半导体两侧产生附加温差将再次产生埃廷斯豪效应,从而产生温差电动势,并叠加在霍尔测量上。为消除上述副效应对霍尔电压测量的影响,在测量时分别改变样品电流<span class="mq-math-mode" latex-data="I_S" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 27.4167px;"><var mathquill-command-id="138">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="134" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="135"><var mathquill-command-id="139">S</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;或磁场B的方向,经推导,得到计算霍尔电压公式<span class="mq-math-mode" latex-data="U_H=\frac{1}{4}\left(U_1-U_2+U_3-U_4\right)" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 388.698px;"><var mathquill-command-id="165">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="142" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="143"><var mathquill-command-id="167">H</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="168">=</span><span class="mq-fraction mq-non-leaf" mathquill-command-id="185" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="186"><span mathquill-command-id="191">1</span></span><span class="mq-denominator" mathquill-block-id="187" style="width: 19.0833px;"><span mathquill-command-id="190">4</span></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-non-leaf" mathquill-command-id="182"><span class="mq-scaled mq-paren" style="transform: scale(1.09, 1.74);">(</span><span class="mq-non-leaf" mathquill-block-id="183"><var mathquill-command-id="169">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="147" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="148"><span mathquill-command-id="170">1</span></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="171">&minus;</span><var mathquill-command-id="172">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="152" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="153"><span mathquill-command-id="173">2</span></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="176">+</span><var mathquill-command-id="177">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="157" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="158"><span mathquill-command-id="178">3</span></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="179">&minus;</span><var mathquill-command-id="180">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="162" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="163"><span mathquill-command-id="181">4</span></span><span style="display: inline-block; width: 0px;">​</span></span></span><span class="mq-scaled mq-paren" style="transform: scale(1.09, 1.74);">)</span></span></span></span>&nbsp;。</p><p><img alt="" height="238" src="/files/testpaper/106/2022/04-07/1922499ab596352893.jpg" width="400" /></p>

实验步骤

<p>测量电路如图所示:</p><p>&nbsp;1.保持磁场(即激励电流<span class="mq-math-mode" latex-data="I_M" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 36.375px;"><var mathquill-command-id="8">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="9">M</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;)大小不变,改变霍尔电流<span class="mq-math-mode" latex-data="I_S" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 27.4167px;"><var mathquill-command-id="8">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="10">S</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;的大小,测绘霍尔电压与电流关系<span class="mq-math-mode" latex-data="U_H-I_S" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 103.635px;"><var mathquill-command-id="17">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="14" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="15"><var mathquill-command-id="18">H</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="19">&minus;</span><var mathquill-command-id="8">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="10">S</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;曲线。取<span class="mq-math-mode" latex-data="I_M" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 36.375px;"><var mathquill-command-id="8">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="21">M</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;=0.45A,电压测量开关选择<span class="mq-math-mode" latex-data="U_H" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 45.0312px;"><var mathquill-command-id="23">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="25">H</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;,分别改变<span class="mq-math-mode" latex-data="I_S" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 27.4167px;"><var mathquill-command-id="26">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="28">S</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;和<span class="mq-math-mode" latex-data="I_M" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 36.375px;"><var mathquill-command-id="26">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="31">M</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;换向开关方向,记录数据。</p><p>&nbsp; 2.保持样品电流<span class="mq-math-mode" latex-data="I_S" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 27.4167px;"><var mathquill-command-id="26">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="33">S</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;不变(<span class="mq-math-mode" latex-data="I_S" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 27.4167px;"><var mathquill-command-id="26">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="33">S</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;=4.0mA),改变励磁电流<span class="mq-math-mode" latex-data="I_M" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 36.375px;"><var mathquill-command-id="26">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="34">M</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;的大小,测量霍尔电压与磁场的关系<span class="mq-math-mode" latex-data="U_H-I_M" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 112.594px;"><var mathquill-command-id="41">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="38" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="39"><var mathquill-command-id="42">H</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="43">&minus;</span><var mathquill-command-id="26">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="34">M</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;曲线。电压测量开关选择<span class="mq-math-mode" latex-data="U_H" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 45.0312px;"><var mathquill-command-id="41">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="38" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="39"><var mathquill-command-id="42">H</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;,分别改变<span class="mq-math-mode" latex-data="I_S" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 27.4167px;"><var mathquill-command-id="44">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="38" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="39"><var mathquill-command-id="45">S</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;和<span class="mq-math-mode" latex-data="I_M" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 36.375px;"><var mathquill-command-id="44">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="38" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="39"><var mathquill-command-id="46">M</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;换向开关方向,记录数据。</p><p>&nbsp; 3.根据测量电路中的电流、磁场、霍尔电压的接线方向及测量数据的正、负,判断本样品的导电类型。</p><p>&nbsp; 4.在零磁场(<span class="mq-math-mode" latex-data="I_M=0" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 86.7917px;"><var mathquill-command-id="44">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="38" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="39"><var mathquill-command-id="46">M</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="47">=</span><span mathquill-command-id="48">0</span></span></span>&nbsp;)下,取<span class="mq-math-mode" latex-data="I_S=0.1mA" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 140.406px;"><var mathquill-command-id="44">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="38" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="39"><var mathquill-command-id="50">S</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="47">=</span><span mathquill-command-id="48">0</span><span mathquill-command-id="51">.</span><span mathquill-command-id="52">1</span><var mathquill-command-id="53">m</var><var mathquill-command-id="54">A</var></span></span>&nbsp;,从电压表单独接线至电流表上,计算该样品的电导率和迁移率。</p><p>&nbsp; 5.观察不等势电位差的影响:不加磁场,逐渐增加霍尔电流<span class="mq-math-mode" latex-data="I_S" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="none" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 27.4167px;"><var mathquill-command-id="44">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="38" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="39"><var mathquill-command-id="50">S</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;,并改变电流的方向,观察电压的变化。</p><p><img alt="" height="279" src="/files/testpaper/106/2022/04-07/19231209bece888001.jpg" width="400" /></p>

5.3 PN结特性

实验目的和仪器

<p><strong>【实验目的】</strong></p><p>1)研究PN结正向压降随温度变化的基本规律;<br />
2)学习用PN结测温的方法;<br />
3)学习一种测量玻耳兹曼常数的方法。</p><p><strong>【实验仪器】</strong></p><p>DH-PN-2型PN结正向特性综合实验仪、DH-SJ温度传感实验装置。</p>

实验原理

<p>1.理想的PN结正向电流<span class="mq-math-mode" latex-data="I_F" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 30.3958px;"><var mathquill-command-id="8">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="9">F</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;和压降<span class="mq-math-mode" latex-data="U_F" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 42.0312px;"><var mathquill-command-id="10">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="9">F</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;之间存在近似关系:<span class="mq-math-mode" latex-data="I_F=I_s\exp\left(\frac{qU_F}{kT}\right)" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 229.552px;"><var mathquill-command-id="11">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="5" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="6"><var mathquill-command-id="9">F</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="12">=</span><var mathquill-command-id="18">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="15" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="16"><var mathquill-command-id="19">s</var></span><span style="display: inline-block; width: 0px;">​</span></span><var class="mq-operator-name" mathquill-command-id="20">e</var><var class="mq-operator-name" mathquill-command-id="21">x</var><var class="mq-operator-name" mathquill-command-id="22">p</var><span class="mq-non-leaf" mathquill-command-id="23"><span class="mq-scaled mq-paren" style="transform: scale(1.2, 2.88);">(</span><span class="mq-non-leaf" mathquill-block-id="24"><span class="mq-fraction mq-non-leaf" mathquill-command-id="26" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="27"><var mathquill-command-id="30">q</var><var mathquill-command-id="36">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="33" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="34"><var mathquill-command-id="37">F</var></span><span style="display: inline-block; width: 0px;">​</span></span></span><span class="mq-denominator" mathquill-block-id="28" style="width: 53.0625px;"><var mathquill-command-id="38">k</var><var mathquill-command-id="39">T</var></span><span style="display: inline-block; width: 0px;">​</span></span></span><span class="mq-scaled mq-paren" style="transform: scale(1.2, 2.88);">)</span></span></span></span>&nbsp;,q为电子电荷;k为玻尔兹曼常数;T为热力学温度;<span class="mq-math-mode" latex-data="I_s" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 24.4375px;"><var mathquill-command-id="18">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="15" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="16"><var mathquill-command-id="19">s</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;为反向饱和电流,可以证明<span class="mq-math-mode" latex-data="I_s=CT^r\exp\left(-\frac{qU_g\left(0\right)}{kT}\right)" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 313.167px;"><var mathquill-command-id="18">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="15" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="16"><var mathquill-command-id="19">s</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="40">=</span><var mathquill-command-id="41">C</var><var mathquill-command-id="48">T</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="45" style="font-size: 26.91px;"><span class="mq-sup" mathquill-block-id="46"><var mathquill-command-id="49">r</var></span></span><var class="mq-operator-name" mathquill-command-id="50">e</var><var class="mq-operator-name" mathquill-command-id="53">x</var><var class="mq-operator-name" mathquill-command-id="54">p</var><span class="mq-non-leaf" mathquill-command-id="55"><span class="mq-scaled mq-paren" style="transform: scale(1.2, 3);">(</span><span class="mq-non-leaf" mathquill-block-id="56"><span mathquill-command-id="58">&minus;</span><span class="mq-fraction mq-non-leaf" mathquill-command-id="59" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="60"><var mathquill-command-id="66">q</var><var mathquill-command-id="72">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="69" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="70"><var mathquill-command-id="73">g</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-non-leaf" mathquill-command-id="74"><span class="mq-scaled mq-paren" style="transform: scale(1, 1.2);">(</span><span class="mq-non-leaf" mathquill-block-id="75"><span mathquill-command-id="77">0</span></span><span class="mq-scaled mq-paren" style="transform: scale(1, 1.2);">)</span></span></span><span class="mq-denominator" mathquill-block-id="61" style="width: 92.7812px;"><var mathquill-command-id="63">k</var><var mathquill-command-id="65">T</var></span><span style="display: inline-block; width: 0px;">​</span></span></span><span class="mq-scaled mq-paren" style="transform: scale(1.2, 3);">)</span></span></span></span>&nbsp;,C是与结面积、掺杂浓度等有关的常数;r是常数(通常取3.4),<span class="mq-math-mode" latex-data="U_g" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span></span><span class="mq-math-mode" latex-data="U_g\left(0\right)" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 86.1562px;"><var mathquill-command-id="72">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="69" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="70"><var mathquill-command-id="73">g</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-non-leaf" mathquill-command-id="80"><span class="mq-scaled mq-paren" style="transform: scale(1, 1.2);">(</span><span class="mq-non-leaf" mathquill-block-id="81"><span mathquill-command-id="83">0</span></span><span class="mq-scaled mq-paren" style="transform: scale(1, 1.2);">)</span></span></span></span>&nbsp;为0K时PN结材料的导带底和价带顶的电势差。整理上述两个式子得到<span class="mq-math-mode" latex-data="U_F=U_g\left(0\right)-\left(\frac{k}{q}\ln\frac{C}{I_F}\right)T-\frac{kT}{q}\ln T^r=U_l+U_{nl}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 617.26px;"><var mathquill-command-id="89">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="86" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="87"><var mathquill-command-id="90">F</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="91">=</span><var mathquill-command-id="97">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="94" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="95"><var mathquill-command-id="98">g</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-non-leaf" mathquill-command-id="99"><span class="mq-scaled mq-paren" style="transform: scale(1, 1.2);">(</span><span class="mq-non-leaf" mathquill-block-id="100"><span mathquill-command-id="102">0</span></span><span class="mq-scaled mq-paren" style="transform: scale(1, 1.2);">)</span></span><span class="mq-binary-operator" mathquill-command-id="103">&minus;</span><span class="mq-non-leaf" mathquill-command-id="126"><span class="mq-scaled mq-paren" style="transform: scale(1.2, 2.88);">(</span><span class="mq-non-leaf" mathquill-block-id="127"><span class="mq-fraction mq-non-leaf" mathquill-command-id="106" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="107"><var mathquill-command-id="110">k</var></span><span class="mq-denominator" mathquill-block-id="108" style="width: 18.8333px;"><var mathquill-command-id="111">q</var></span><span style="display: inline-block; width: 0px;">​</span></span><var class="mq-operator-name" mathquill-command-id="112">l</var><var class="mq-operator-name" mathquill-command-id="113">n</var><span class="mq-fraction mq-non-leaf" mathquill-command-id="114" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="115"><var mathquill-command-id="118">C</var></span><span class="mq-denominator" mathquill-block-id="116" style="width: 29.1354px;"><var mathquill-command-id="124">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="121" style="font-size: 24.219px;"><span class="mq-sub" mathquill-block-id="122"><var mathquill-command-id="125">F</var></span><span style="display: inline-block; width: 0px;">​</span></span></span><span style="display: inline-block; width: 0px;">​</span></span></span><span class="mq-scaled mq-paren" style="transform: scale(1.2, 2.88);">)</span></span><var mathquill-command-id="129">T</var><span class="mq-binary-operator" mathquill-command-id="130">&minus;</span><span class="mq-fraction mq-non-leaf" mathquill-command-id="131" style="font-size: 26.91px;"><span class="mq-numerator" mathquill-block-id="132"><var mathquill-command-id="136">k</var><var mathquill-command-id="137">T</var></span><span class="mq-denominator" mathquill-block-id="133" style="width: 32.2917px;"><var mathquill-command-id="135">q</var></span><span style="display: inline-block; width: 0px;">​</span></span><var class="mq-operator-name" mathquill-command-id="138">l</var><var class="mq-operator-name mq-last" mathquill-command-id="139">n</var><var mathquill-command-id="145">T</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="142" style="font-size: 26.91px;"><span class="mq-sup" mathquill-block-id="143"><var mathquill-command-id="146">r</var></span></span><span class="mq-binary-operator" mathquill-command-id="147">=</span><var mathquill-command-id="158">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="150" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="151"><var mathquill-command-id="160">l</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="163">+</span><var mathquill-command-id="159">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="155" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="156"><var mathquill-command-id="161">n</var><var mathquill-command-id="162">l</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;。这就是PN结正向压降作为电流和温度函数的表达式,是PN结温度传感器的基本方程。令<span class="mq-math-mode" latex-data="I_F" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 30.3958px;"><var mathquill-command-id="169">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="166" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="167"><var mathquill-command-id="170">F</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;=常数,则正向压降只随温度变化,但在方程中,包含线性和非线性两项。</p><p>&nbsp; 2.对于杂质全部电离、本征激发可忽略的温度区间,根据对<span class="mq-math-mode" latex-data="U_{nl}" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 46.5312px;"><var mathquill-command-id="171">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="166" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="167"><var mathquill-command-id="172">n</var><var mathquill-command-id="173">l</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;项所引起的线性误差的分析可知,在恒流供电条件下,PN结的<span class="mq-math-mode" latex-data="U_F" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 42.0312px;"><var mathquill-command-id="171">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="166" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="167"><var mathquill-command-id="174">F</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;对T的依赖关系主要取决于线性项<span class="mq-math-mode" latex-data="U_l" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 33.0729px;"><var mathquill-command-id="171">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="166" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="167"><var mathquill-command-id="176">l</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;,即正向压降几乎随温度升高而线性下降,这就是PN结测温的依据。<span class="mq-math-mode" latex-data="U_F-T" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 93.8542px;"><var mathquill-command-id="171">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="166" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="167"><var mathquill-command-id="178">F</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="180">&minus;</span><var mathquill-command-id="181">T</var></span></span>&nbsp;特性还因PN结的材料而异,对于宽带材料的PN结,其高温端的线性区宽;而材料杂质电力能小的PN结,则低温端的线性区宽。对于给定的PN结,即使在杂质导电和非本征激发温度范围内,其线性度亦随温度的高低而有所不同,这是非线性项引起的。<span class="mq-math-mode" latex-data="U_F-T" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 59.3021px;"><var mathquill-command-id="171">U</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="166" style="font-size: 16.56px;"><span class="mq-sub" mathquill-block-id="167"><var mathquill-command-id="178">F</var></span><span style="display: inline-block; width: 0px;">​</span></span><span class="mq-binary-operator" mathquill-command-id="180">&minus;</span><var mathquill-command-id="181">T</var></span></span>&nbsp;的线性度在高温端由于低温端,这是PN结温度传感器的普遍规律。此外,减小<span class="mq-math-mode" latex-data="I_F" style="font-size: 29.9px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 30.3958px;"><var mathquill-command-id="182">I</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="166" style="font-size: 26.91px;"><span class="mq-sub" mathquill-block-id="167"><var mathquill-command-id="178">F</var></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;也可以改变线性度,但并不能从根本上解决问题。</p><p>&nbsp; 将PN结测温电路与恒流、放大等电路集成一体,便构成集成电路温度传感器。</p>

实验步骤

<p><strong>1.实验系统检查与连接</strong><br />
实验前,请参照仪器使用说明,将DH-SJ型温度传感器实验装置上的&ldquo;加热电流&rdquo;开关置&ldquo;关&rdquo;位置,将&ldquo;风扇电流&rdquo;开关置&ldquo;关&rdquo;位置,接上加热电源线。插好Pt100温度传感器和PN结温度传感器,两者连接均为直插式。PN结引出线分别插入PN结正向特性综合试验仪上的+V、-V和+I、-I。注意插头的颜色和插孔的位置。<br />
打开电源开关,温度传感器实验装置上将显示出室温Tg,记录下起始温度TR。</p><p><strong>2.玻耳兹曼常数k的测定</strong><br />
(1)PN结I-U关系的测量&nbsp; PN结正向电流随正向电压按指数规律变化。若能测得 PN结I-U关系,则可求出玻耳兹曼常数A。请根据所需数据的测量要求,自己设计实验过程、确定实验条件,将PN结正向特性综合试验仪上的电流量程置于适当档位,调整电流调节旋钮以改变正向电流I,输出示值,观察记录相应的正向电压U值读数,将记录数据填入表。<br />
&nbsp;( 2 )选择合适的数据处理方法,确定玻尔兹曼常数,将所得结果与标准值k=1.3807&times;10-23J/K相比。</p><p><strong>3.至少完成对一种PN结材料的U-T曲线的测量及数据处理</strong><br />
&nbsp; &nbsp;(1)测定U-T关系曲线 选择合适的正向电流Ip(如Iy=50&mu;A,一般选小于100&mu;A的值,以减小自身热效应),并保持不变。实验的起始温度T。可直接取为室温TR,记录相应的Ur(TR)。将DH-SJ型温度传感器实验装置上的&ldquo;加热电流&rdquo;开关置&ldquo;开&rdquo;位置,根据目标温度,选择合适的加热电流(在实验时间允许的情况下,加热电流可以取得小一点,如0.3~0.6A之间)。这时加热炉内温度开始升高,记录对应的Ug和T填入表。可按Up每改变10mV或5mV读取一组U、T值,这样可以减小测量误差。<br />
注意:在整个实验过程中,正向电流e应保持不变。升温速率要慢,且设定的温度不宜过高(硅管必须控制在120℃以内,错管必须控制在45℃以内)。<br />
根据表格的数据,绘制U-T曲线。</p><p>&nbsp; &nbsp;(2)求被测PN结正向压降随温度变化的灵敏度S(mV/℃) 作出的U-T曲线斜率就是S。<br />
&nbsp; &nbsp;(3)估算被测PN结材料的禁带宽度E。(0)=qU(0) 忽略非线性项的影响,U(0)=(0℃)<br />
Up(0℃)十dUpoT △T=U (0℃)+S&middot;△T。此时温差△T=-273.2K。将根据此式算得到的OK时的禁带宽度E(0)值与公认值(硅E(0)=1.21eV,锗E(0)=0.78eV)比较,求其相对误差。</p>

5.4 液体变温粘滞系数测量及PID使用

实验目的和仪器

<p><strong>【实验目的】</strong></p><p>1)用落球法测量不同温度下蓖麻油的黏度。<br />
2)了解PID调节,观察实验装置的PID控温行为。</p><p><strong>【实验仪器】</strong></p><p>落球法变温黏滞系数实验仪、ZKY-PID温控实验仪、停表、千分尺、小钢球若干等。</p>

实验原理

【5.4 实验原理部分缺失】

实验步骤

<p>实验内容为研究蓖麻油黏度随温度的变化。在室温至55℃温度范围,至少取4个同温度,测量蓖麻油黏度。具体实验步骤如下:<br />
1)检查温控仪面板上的水位管,将水箱水加到适当值:平常加水从仪器顶部的注水孔注入。若水箱排空后第一次加水,应该用软管从出水孔将水经水泵加入水箱,以便排出水泵内的空气,避免水泵空转(无循环水流出)或发出嗡鸣声。<br />
2)设定PID参数:本仪器的PID的参数已经是通过理论分析和大量的实验得到的一个最符合本仪器的参数,已经达到最佳控制,故不可调节。<br />
3)测定小球直径:当液体黏度及小球密度一定时,雷诺数Re xdP。在测量蓖麻油的黏度时采用了直径1~2mm的小球,这样可不考虑雷诺修正或只考虑1级雷诺修正。<br />
用千分尺测定小球的直径d,将数据记入表中。<br />
4)测定小球在液体中下落的速度并计算黏度<br />
温控仪温度达到设定值后再等约10min,使样品管中的待测液体温度与加热水温完全一致,才能测液体黏度。<br />
借助导向管,用挖油勺盛住小球沿样品管中心轻轻放入液体,观察小球是否一直沿中心下落,若样品管倾斜,应调节其铅直。测量过程中,尽量避免对液体的扰动。<br />
用停表测量小球落下一段距离的时间t,并计算小球速度to,用公式计算黏度n,记入表中。<br />
用表中n的测量值作图,给出黏度随温度的变化关系。<br />
实验全部完成后,用磁铁将小球吸引至样品管口,取出放入小钢球盒中保存,以备下次实验使用。<br />
注意:实验结束后,要将仪器及各个配件复位,将实验台收拾整洁。</p>

5.6 迈克尔逊干涉仪

实验目的和仪器

<p><strong>【实验目的】</strong></p><p>1)学会使用迈克耳孙干涉仪。<br />
2)观察等倾、等厚和非定域干涉现象。<br />
3)测量氦氛激光的波长、钠黄光谱线的波长和钠光双线的波长差。</p><p><strong>【实验仪器】</strong><br />
氦氛激光光源、钠光灯、迈克耳孙干涉仪、毛玻璃屏</p>

实验原理

<p><strong>1.迈克耳孙干涉仪的原理</strong><br />
&nbsp; &nbsp; &nbsp;迈克耳孙干涉仪的光路图如图所示,光源 S 出发的光经过呈45&deg;放置的背面镀银的半透玻璃板 P1(镀有银的表面在图中用加粗的线来表示)被分成相互垂直的强度几乎相等的两東光,光路 1 通过 M1 镜反射并再次通过 P1照射在观察平面E上,光路 2 通过厚度、折射率与 P1 相同的玻璃板 P2后由 M2镜反射再次通过 P2并由 P1背面的反射层反射照射在观察平面E上。图中平行于 M1的 M2&#39;是 M2经 P1 反射所成的虚像,即 P1到 M2与 P1到 M2&rsquo;的光程距离相等,故从 P1到 M2的光路可用 P1 到 M2&#39;等价代替。这样可以认为 M1与M2&#39;之间形成了一个空气间隙,这个空气间隙的厚度可以通过移动 M1完成,空气间隙的夹角可以通过改变 M1镜或 M2镜的角度实现。当 M1与 M2&#39;平行时可以在观察平面E处观察到等倾干涉现象,当 M1与 M2&lsquo;有一定的夹角时可以在观察平面E处观察到等厚干涉现象。</p><p><img alt="" height="534" src="/files/testpaper/106/2022/03-29/1720320c37bb281382.jpg" width="400" /></p><p><strong>2.干涉条纹的形成原理</strong><br />
(1)等倾干涉条纹调节反光镜 M1 与 M2,使之相互垂直,即此时 M1与 M2&#39;相互行,在 M1与M2&#39;之间形成了一个厚度不变的空气间隙,在扩展光源的情况下,由于入射角相同的光经空气层上下表面反射形成的反射光在相遇点有相同的光程差,也就是说,只要是入射角日相同的光就形成同一条纹,故这些倾角不同的光束经空气层反射后形成的干涉图样是一些明暗相间的同心圆环。这种干涉称为等倾干涉。等倾干涉条纹只呈现在透镜的焦平面上,不用透镜时产生的干涉条纹应在无限远处,所以我们说等倾于涉条纹定域于无穷远处。人的眼睛相当于一个可成像的透镜,这时用眼睛看无穷远处的干涉条纹可以观察到一系列的同心圆,涉条纹的级次最高,当d变化时,可看到条纹的&ldquo;吐出&rdquo;或&ldquo;吞没&rdquo;。</p><p>等倾干涉条纹的特点:</p><p>1)圆心处,&theta;=0,光程差△=2d,此时光程差最大,条纹级次最高,当d变化时,可看到条纹的&ldquo;吐出&rdquo;或&ldquo;吞没&rdquo;。即:圆心处对于j级亮条纹来说,满足 2dcos &theta;=j&lambda;。当移动 M1增大d时,&theta;将随之增大,即该级亮环的半径增大。这样,将观察到干涉条纹不断从中心&ldquo;吐出&rdquo;并向外扩大。相反,当d减小时,条纹圆环将不断向中心收缩,中心点的干涉级变小,看上去圆环好像被一个个&ldquo;吞没&rdquo;。若在移动 M1的过程中,有N个条纹被&ldquo;吞进&rdquo;或被&ldquo;吐出&rdquo;(或者有N个条纹通过某一个参考点),则 M1移动的距离为&Delta;d=N&lambda;/2</p><p>2)等倾干涉条纹的条纹间距随着 M1与 M2间距离d的增大而变密</p><p>第j级和j+1级亮纹满足公式:</p><p>2dcos<span class="mq-math-mode" latex-data="\Theta" style="font-size: 18.4px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 16.9792px;"><span mathquill-command-id="3">&Theta;</span></span></span>&nbsp;j=j<span class="mq-math-mode" latex-data="\Theta\lambda" style="font-size: 18.4px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 12.9271px;"><span class="mq-nonSymbola" mathquill-command-id="10">&lambda;</span></span></span>&nbsp;</p><p>2dcos<span class="mq-math-mode" latex-data="\Theta" style="font-size: 18.4px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 16.9792px;"><span mathquill-command-id="3">&Theta;</span></span></span>&nbsp;j+1=(j+1)<span class="mq-math-mode" latex-data="\Theta\lambda" style="font-size: 18.4px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 12.9271px;"><span class="mq-nonSymbola" mathquill-command-id="10">&lambda;</span></span></span>&nbsp;</p><p>等倾干涉圆环具有中心疏,边缘密的特点。</p>

实验步骤

<p><strong>(1)观察非定域干涉条纹</strong></p><p>1)通过粗调手轮打开激光光源,调节激光器使其光束大致垂直于平面反光镜 M2入射,并取掉投影屏E,可以看到两排激光点,<br />
2)通过粗调手轮移动 M,镜的位置,使得通过分光板分开的两路光光程大致相等。<br />
3)调节 MI、M2镜后面的两个旋钮(注意要均匀调整),使两排激光点重合为一排,并使两个最亮的光点重合在一起。此时再放上投影犀 E,就可以观察到干涉条纹。<br />
4)仔细调节 M1、M2,镜后面的两个旋钮,便 M 与 Mg/平行,这时在屏上可以看到同心圆条纹,这些条纹为非定域条纹,当移动投影屏 E 位置或改变两路光的光程差时,圆环将变大或缩小。<br />
5)转动微调(或粗调)手轮,观察干涉条纹的形状、疏密及中心&ldquo;吞&rdquo;&ldquo;吐&rdquo;条纹随光程差改变的变化情况。向同一个方向缓慢旋动微调手轮,当观察到条纹显著的涌出或缩进时,开始记录此时持续沿同一方向旋动微调手轮,条纹中心每&ldquo;吞&rdquo;或&ldquo;吐&rdquo;50 条条纹</p><p><strong>(2)测量 He-Ne 激光波长</strong></p><p>向同一个方向缓慢旋动微调手轮,当观察到条纹显著的涌出或缩进时,开始记录此时M1的位置m0,持续沿同一方向旋动微调手轮,条纹中心每&ldquo;吞&rdquo;或&ldquo;吐&rdquo;50 条条纹记一次h值。连续记录 10 次,并将分别记录的m0,m1,m2,&hellip;,m9记录。利用最小二乘法求出 &Delta;m,并计算出激光波长入及其误差。</p><p><strong>(3)&nbsp;测量钠黄光波长</strong></p><p>1)将光源换为钠灯,在钠光灯与里之间放置一块壅砂玻璃,并将投影屏B取下,通过分光板A直接观察干涉条纹/<br />
2)缓慢旋动微调手轮,观察钠灯产生的干涉条纹的吞吐。<br />
3)每&ldquo;吞&rdquo;或&ldquo;吐&rdquo;25 条条纹记一次 M 镜的位置r值,连续记录6次,并将分别记录的r0,r1,,&hellip;, r5记录在表 。利用最小二乘法求出了,并计算出波长&Delta;r及其误差。</p><p><strong>(4)钠光双线波长差 △&lambda;的测定</strong></p><p>继续缓慢旋动微调手轮移动 M1镜,观察到条纹的反衬度周期变化,记录条纹从反衬度变到极小的位置到下一个极小的位置时 M1镜对应的位置d值,连续记录6 次,并将分别记录的d1,d2,&hellip;,d6记录在表中,利用最小乘法求出 △d,计算钠光双线的波长差及其误差。</p>

5.7 全息照相

实验目的和仪器

<p><strong>实验目的:</strong></p><p>(1)了解全息照相的基本原理;<br />
(2)掌握全息照相方法及底片冲洗方法;<br />
(3)观察物像再现。</p><p><strong>实验仪器:</strong></p><p>激光器、成套全息照相光具元件及隔震光学平台、白屏(用以接收光和观察干涉条纹图样)</p><p>硅光电池及电压表、全息干板、被照物体、显影液和定影液等。</p>

实验原理

<p>普通照相是通过照相机透镜把来自物体的光聚焦于底板上,底板经过曝光冲洗后,我们就得到了物体的负片,负片上各点的沉积透光性与物体上各点的光强一一对应,我们在底板上看到的是物体的负向。由于底板只是记录了物体上各点的光强分布,并没有记录反映物体之间相对位置、远近的相位信息,因而是所拍摄物体的二维平面像。全息照相记录的是整个物体发出的光波(及物体上各点发出的光波的叠加)。借助于参考光,全息照相术用干涉的方法记录了物光波的振幅和相位分布,即记录下物光波与参考光波相干后的全部信息。这些信息以干涉条纹花样的形式被记录,可以说全息照片就是储存起来的被拍摄物体的干涉图样。所以,全息底片上显现的不是物体的影像,而是细密的干涉条纹,就好像是一个复杂的衍射光栅。若再想看到物体的影像,必须对全息底片进行适当的再照明,重建原来的物光波,才能再现物体的三维立体像。</p><p><strong>(1)全息记录</strong></p><p>全息记录的光路图如图所示。</p><p><img alt="" height="185" src="/files/testpaper/106/2022/03-15/204414eec0b1530729.jpg" width="400" /></p><p>用激光照射物体,物体因漫反射而发出物光波。波场每一点的振幅和相位都是空间坐标的函数。我们用O表示物体每一点发出的物光的复振幅与相位。用同一激光束经分束镜分出的另一部分光直接照射到底板上,这个光波称为参考光波,它的振幅和相位也是空间坐标的函数,其复振幅和相位用 R 表示,参考光通常为平面或球面波。这样在记录光信息的底板上的总光场是物光与参考光的叠加。</p><p>在底板上光强I的分布就是物光波和参考光波干涉形成的干涉条纹,我们用感光底板将其记录下来。由于干涉条纹在底板上各点的强度取决于物光波和参考光波在各点的振幅和相位,因而底板上就保留了物光波的振幅和相位分布的信息。</p><p><strong>(2)物像再现</strong></p><p>底板经过曝光冲洗以后,形成各处透光率不相同的全息照片,它相当于一个复杂的光栅。一般来说,光透过这样的全息照片时,振幅及相位都要发生变化。如果令&nbsp; &nbsp;t =透过光的复振幅/入射光的复振幅<br />
则复振幅透过率 t 一般为复数。但对于平面吸收型全息照片(这种底板各处透过率的不同仅仅是由于沉积的银层对光的不同吸收引起的),t 为实数。如果曝光及冲洗合适(线性处理),可使得&nbsp;&nbsp;t=t<sub>0</sub>-KI<br />
式中, t<sub>0</sub>为未曝光部分的透过率; t<sub>0&nbsp;</sub>和 K 都是常数。</p><p>物像的再现是用光照射已经摄制好的全息照片并观察透过光,这个过程称为波前重现。</p><p>由于底片上任何一小部分都包含整个物体的信息,所以只利用拍摄的全息底片的一小部分也能再现整个物像。</p><p><strong>(3)全息照片拍摄要素</strong></p><div>1.&nbsp; &nbsp;相干光源&nbsp; &nbsp; &nbsp;</div><div>物光和参考光必须是相干光,利用分束镜把激光器发出的光分成两束,一束作为物光,一束作为参考光,并使其光程大致相等,以使两束光能够发生相干干涉。</div><div>2.&nbsp; &nbsp;避免系统震动</div><div>物光和参考光形成的干涉条纹间距很小,每毫米有上千条条纹。在曝光过程(曝光时间10s~20s)中,如果条纹移动超过半个条纹的宽度,就不能形成全息图;条纹移动小于半个条纹宽度,全息图像虽然可以形成,但清晰度会受到影响。因此,在曝光过程中必须保持干涉条纹稳定不变。为此,光源、光路中各光学元件、被摄物体和感光底板都必须放在防震平台上,使外界各种微小振动不致干扰条纹的记录。</div><div>3.&nbsp; &nbsp;物光、参考光的强度</div><div>从干涉理论知,当物光与参考光在干板处强度相等时,干涉条纹的反衬度最好。在实际光路中调节光路时,一般尽可能使全息干板处物光光强尽可能大,适当减弱参考光的光强,使物光与参考光的光强比为1:3到1:5。</div><div>4.&nbsp; &nbsp;&nbsp;全息记录干板</div><div>全息记录干板是表面涂有一层感光乳胶的玻璃,这是由于玻璃有一定刚度,不会变形弯曲。对于不同波长的激光,要选用不同型号的感光乳胶。</div><div>物光与参考光夹角越大,条纹的间距就越小,对干板的分辨率要求就越高,所以夹角不宜太大。但另一方面,夹角大,物象再现时可在较大范围内从不同角度观察物象,反之则观察窗较小。一般使物光与参考光之间的夹角控制在20&deg;~50&deg;范围内。</div>

实验步骤

<p><strong>一、全息图拍摄</strong><br />
1.&nbsp; &nbsp; 按图所示配置光路系统。</p><p><img alt="" height="185" src="/files/testpaper/106/2022/03-15/204414eec0b1530729.jpg" width="400" /></p><p>光路系统应满足下列条件:<br />
(1)物光束和参考光束由分束镜至感光板之间的光程应大致相等。<br />
(2)用扩束镜将物光束扩展到使整个被摄物都能受到光照,参考光束也应扩展使感光板有均匀的光照。注意全息干板与被摄物的距离应控制在 5cm 之内,且应保证全息干板尽可能正对被摄物,以接收多的物光,参考光应均匀照明并覆盖整块全息干板。<br />
(3)照在感光板上的物光束和参考光束之间的夹角在20&deg;~50&deg;范围为宜。<br />
(4)关闭室内照明灯,在放感光板的地方,物光与参考光的光强比为1:3到1:5范围为宜。</p><p>2.&nbsp; &nbsp; 将激光器出射的激光挡住,装夹好全息干板,使乳胶面向着被拍摄物体。激光曝光10~20s,特别要注意在曝光过程中绝对不要触及防震台并保持室内安静。</p><p>3.&nbsp; &nbsp; 显影和定影:按显影-水洗-定影-水洗-晾干的程序处理。</p><p><strong>二、物像再现</strong></p><p>1.&nbsp; &nbsp; 将晾干的全息图片放回原位置,感光乳胶面仍向着物体,用原参考光照射,去掉物,在全息图的后方观察,即可看到位置、大小与原物一样的三维立体像。试改变观察角度,参考光源到干板的距离,观察物像的变化。</p><p>2.&nbsp; &nbsp; 挡掉全息图的一部分,仍然可看到完整的物体像。</p>

5.8 用光栅测量光波波长

实验目的和仪器

<p><strong>【实验目的】</strong></p><p>(1)学习调节和使用分光仪观察光栅衍射的现象。</p><p>(2)学习利用光栅衍射测量光波波长的原理和方法。</p><p>(3)了解角色散与分辨本领的意义以及测量方法。</p><p><strong>【实验仪器】</strong></p><p>JJY分光仪(1&#39;)、光栅、平行平面反射镜、汞灯等。</p>

实验原理

<p><strong>1.光栅方程</strong><br />
光栅是一种重要的分光元件,分为透射光栅和反射光栅。本实验中我们使用的是透射光栅。在一块透明的平板上刻有大量相互平行等宽等间距的刻痕,这样一块平板就是一种透射光栅,其中刻痕部分为不透光部分。</p><p>若刻痕之间透光部分(即狭缝)的宽度为 a ,刻痕宽度为 b ,则光栅常数为 d = a + b 。通常,光栅常数是很小的,例如,在10mm内刻有3000条等宽等间距的狭缝。<br />
当一束波长为入的平行光垂直照射在光栅上时(如图所示)。</p><p><img alt="" height="182" src="/files/testpaper/106/2022/03-17/191835b3e02d671930.jpg" width="300" /></p><p>每一个狭缝透过的光都要发生衍射,向各个方向传播。经过光栅衍射,与光栅面法线成<span class="mq-math-mode" latex-data="\Phi"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><span mathquill-command-id="3">&nbsp;&Phi;&nbsp;</span></span></span>角的平行光,经透镜后会聚于透镜焦平面处屏上一点 P<sub>1</sub> ,<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Phi;&nbsp;</span>角称为衍射角。由于光栅上各狭缝是等间距的,所以沿&nbsp;<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Phi;&nbsp;</span>角方向的相邻光束间的光程差都等于 dsin<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Phi;</span> ,因为光程差一定,它们彼此之间将发生干涉。用透镜将经过光栅衍射的平行光会聚于透镜焦平面处屏上,将呈现由单缝衍射和多缝干涉综合效果所形成的光栅衍射条纹。</p><p>当沿<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Phi;</span>角方向传播的相邻光束间光程差 dsin<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Phi;</span>&nbsp;等于入射光波长的整数倍时,各缝射出的、聚焦于屏上 P<sub>1</sub> 点的光因相干叠加得到加强,形成明条纹。因此,光栅衍射明纹的条件是<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Phi;</span>必须满足&nbsp; &nbsp;&nbsp;dsin<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Phi;=k</span><span class="mq-math-mode" latex-data="\lambda"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><span class="mq-nonSymbola" mathquill-command-id="5">&lambda;</span></span></span><span>&nbsp;</span>&nbsp; (k=0,<span class="mq-math-mode" latex-data="\pm"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><span mathquill-command-id="7">&plusmn;</span></span></span><span>&nbsp;1,</span><span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&plusmn; 2&hellip;&hellip;</span>)</p><p>此式称为光栅方程,它是研究光栅衍射的基本公式。</p><p>满足光栅方程的明条纹称为主极大条纹,也称为光谱线,k称为主极大级数。k=0时,<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Phi;=0,</span>称为零级条纹,对应于中央明条纹,中央明条纹聚集了入射光总能量的大部分,因而较亮。k=<span class="mq-math-mode" latex-data="\pm" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 18.3px;"><span mathquill-command-id="7">&plusmn;</span></span></span>&nbsp;1,k=<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&plusmn; 2,&hellip;&hellip;</span>分别为对称地分布在中央明条纹两侧的第1级、第2级&hellip;主极大条纹。</p><p>由|sin<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Phi;</span>|<span class="mq-math-mode" latex-data="\le"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><span class="mq-binary-operator" mathquill-command-id="9">&le;</span></span></span><span>&nbsp;</span>1知,主极大的级数限制是&nbsp; &nbsp;k<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&le;d/</span><span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; white-space: nowrap;">&lambda;</span></p><p>即在给定光栅常数的情况下,光谱级数是有限的,再综合考虑不同级次谱线光强的变化,实验中通常最多能观察到第2、第3级主极大条纹。</p><p>用分光仪测得第 k 级谱线的衍射角后,若已知光栅常数 d 就可求出入射光的波长,这就是用光栅测量光波波长的基本原理。反之,若给定波长<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; white-space: nowrap;">&lambda;</span>,则可求出光栅常数 d 。</p><p>如若单色平行光不是垂直照射在光栅上,而是以入射角<span class="mq-math-mode" latex-data="\Theta"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><span mathquill-command-id="11">&Theta;</span></span></span><span>&nbsp;</span>斜入射到光栅上,则光栅方程变为<br />
&nbsp;d (sin<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Theta;</span>+ sin Ф)= k&nbsp;<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; white-space: nowrap;">&lambda;</span>( k =0,&plusmn; 1 ,&plusmn;2,&hellip;&hellip;)</p><p><strong>2.光栅色散本领与分辨本领</strong></p><p>由式&nbsp; dsin<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Phi;=k</span><span class="mq-math-mode" latex-data="\lambda" style="font-size: 18.4px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 12.925px;"><span class="mq-nonSymbola" mathquill-command-id="5">&lambda;</span></span></span>&nbsp;&nbsp; (k=0,<span class="mq-math-mode" latex-data="\pm" style="font-size: 18.4px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 18.3px;"><span mathquill-command-id="7">&plusmn;</span></span></span>&nbsp;1,<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&plusmn; 2&hellip;&hellip;</span>) 可知,如果入射光波长不同,则同等级光谐衍射角不同,波长越长,衍射角越大,这就是光栅的分光原理。如果入射光是复色光,则由于波长不同,衍射角&nbsp;<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Phi;&nbsp;</span>也各不同,于是不同的波长就被分开,按波长从小到大依次排列,成为一组彩色条纹,这就是光谱,这种现象称为色散现象。</p><p>角色散率表示光栅将不同波长的同级谱线分开的程度,是光栅的主要性能参数之一。</p><p>设两单色光波波长分别为<span class="mq-math-mode" latex-data="\lambda"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><span class="mq-nonSymbola" mathquill-command-id="3">&lambda;</span></span></span><sub>1</sub>、<span class="mq-math-mode" latex-data="\lambda" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 12.925px;"><span class="mq-nonSymbola" mathquill-command-id="3">&lambda;</span></span></span><sub>2</sub>,其波长差为<span class="mq-math-mode" latex-data="\delta"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><var mathquill-command-id="5">&delta;</var></span></span><sub><span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; white-space: nowrap;">&lambda;</span></sub><span>=| </span><span class="mq-math-mode" latex-data="\lambda" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 12.925px;"><span class="mq-nonSymbola" mathquill-command-id="3">&lambda;</span></span></span><sub>1</sub>-<span class="mq-math-mode" latex-data="\lambda" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 12.925px;"><span class="mq-nonSymbola" mathquill-command-id="3">&lambda;</span></span></span><sub>2</sub><span>|,第k级衍射角之差为</span><span class="mq-math-mode" latex-data="\delta" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 12.5625px;"><var mathquill-command-id="5">&delta;</var></span></span><sub><span class="mq-math-mode" latex-data="\Phi"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><span mathquill-command-id="7">&Phi;</span></span></span></sub><span>&nbsp;,则该级次的角色散率为&nbsp; &nbsp; D</span><span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 15.3333px; white-space: nowrap;"><sub>&Phi;</sub>=</span><span class="mq-math-mode" latex-data="\delta" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 12.5625px;"><var mathquill-command-id="5">&delta;</var></span></span><sub><span class="mq-math-mode" latex-data="\Phi" style="font-size: 15.3333px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 14.4375px;"><span mathquill-command-id="7">&Phi;</span></span></span></sub><span class="mq-math-mode" latex-data="\Phi" style="font-size: 15.3333px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 14.4375px;"><span mathquill-command-id="7">/</span></span></span><span class="mq-math-mode" latex-data="\delta" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 12.5625px;"><var mathquill-command-id="5">&delta;</var></span></span><sub><span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; white-space: nowrap;">&lambda;</span></sub></p><p>式中,&nbsp;<span class="mq-math-mode" latex-data="\delta" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 12.5625px;"><var mathquill-command-id="5">&delta;</var></span></span><sub><span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; white-space: nowrap;">&lambda;&nbsp;</span></sub>的单位为mm,<span class="mq-math-mode" latex-data="\delta" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 12.5625px;"><var mathquill-command-id="5">&delta;</var></span></span><sub><span class="mq-math-mode" latex-data="\Phi" style="font-size: 15.3333px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 14.4375px;"><span mathquill-command-id="7">&Phi;</span></span></span></sub>的单位为弧度。由式 dsin<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Phi;=k</span><span class="mq-math-mode" latex-data="\lambda" style="font-size: 18.4px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 12.925px;"><span class="mq-nonSymbola" mathquill-command-id="5">&lambda;</span></span></span>&nbsp;得 D<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 15.3333px; white-space: nowrap;"><sub>&Phi;</sub>=</span><span class="mq-math-mode" latex-data="\delta" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 12.5625px;"><var mathquill-command-id="5">&delta;</var></span></span><sub><span class="mq-math-mode" latex-data="\Phi" style="font-size: 15.3333px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 14.4375px;"><span mathquill-command-id="7">&Phi;</span></span></span></sub><span class="mq-math-mode" latex-data="\Phi" style="font-size: 15.3333px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 14.4375px;"><span mathquill-command-id="7">/</span></span></span><span class="mq-math-mode" latex-data="\delta" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 12.5625px;"><var mathquill-command-id="5">&delta;</var></span></span><sub><span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; white-space: nowrap;">&lambda;</span></sub><span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; white-space: nowrap;">=k/dcos</span>Ф&nbsp;,该式表明角色散率 D<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 15.3333px; white-space: nowrap;"><sub>&Phi;&nbsp;</sub></span>与光栅常数成反比,与谱线级数k成正比,谱线级次越高,角色散率越大。</p><p>光栅分辨本领是光栅的另一重要性能指标,比色散率更具实际意义。有了大的色散,并不能保证能分辨出两条靠近的谱线,而这一性能是由分辨本领来表征的。分辨本领定义为刚好能分辨开的两条单色谱线的波长差&nbsp;<span class="mq-math-mode" latex-data="\delta" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 12.5625px;"><var mathquill-command-id="5">&delta;</var></span></span><sub><span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; white-space: nowrap;">&lambda;&nbsp;</span></sub>与这两种波长的平均值之比。</p>

实验步骤

<p>光栅方程&nbsp;dsin<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Phi;=k</span><span class="mq-math-mode" latex-data="\lambda" style="font-size: 18.4px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 12.925px;"><span class="mq-nonSymbola" mathquill-command-id="5">&lambda;</span></span></span>&nbsp;是在平行光垂直入射到光栅平面的条件下得出的,因此,在实验中调节仪器时要注意满足此要求。</p><p>以下是调节的具体步骤:</p><p><strong>(1)</strong>按实验4.14【实验装置】部分的&ldquo;1.分光仪的构造&ldquo;和&ldquo;2.分光仪的调节&rdquo;内容调节好分光仪。</p><p><strong>(2)</strong>调节光栅平面使之与平行光管光轴垂直<br />
分光仪调节好后,将光栅按如图所示的方式放置物台上。</p><p><img alt="" height="206" src="/files/testpaper/106/2022/03-17/1917106d4171445937.jpg" width="300" /></p><p>挡住光源的光,开亮望远镜上的小灯,转动载物台(连同光栅),从望远镜中观察光栅平面反射回来的绿色十字像,如像的水平线不在分划板上十字线水平线处,只能调节载物台水平调节螺钉20的 B<sub>2</sub>(或 B<sub>3</sub>),使二者重合,此时光栅平面已垂直于平行光管光轴。</p><p><strong>(3)</strong>调节光栅使其透光狭条与仪器主轴平行:转动望远镜观察衍射条纹的分布。以分划板横线为参考线,比较左右两侧谱线在视场中的高度是否一致。如果高度不同,说明光栅狭缝与仪器主轴不平行,可调节载物台螺钉 B<sub>1</sub> ,(千万不能动 B<sub>2</sub> 和 B<sub>3</sub> )使谱线等高。</p><p>经过上述调节以后,就可以进行衍射角的测量了。</p><p><strong>(4)</strong>用汞灯照亮平行光管的狭缝,使平行光垂直照射在光栅上,转动望远镜定性观察谱线的分布规律与特征;然后改变平行光在光栅上的入射角度,转动望远镜定性观察谱线的分布的变化。</p><p><strong>(5)</strong>测量肉眼可以很清楚看到的汞灯蓝色、绿色、黄色I、黄色II 4条谱线。使平行光垂直照射在光栅上,先使望远镜对准中央亮线,然后向左转动望远镜,对观察到的每一条汞光谱线,使谱线中央与分划板的垂直线重合,将望远镜此时的角位置(&nbsp;P<sub>左</sub>,P<sup>&lsquo;</sup><sub>左</sub> )记录到数据表中。向右转动望远镜进行测量,得到各谱线的( P<sub>右</sub>,&nbsp;P<sup>&rsquo;</sup><sub>右</sub>),记录到数据表中。</p>

5.9 光栅光谱仪的使用

实验目的和仪器

<p><strong>【实验目的】</strong></p><p>(1)了解平面反射式闪耀光栅的分光原理及主要特性。</p><p>(2)了解光栅光谱仪的结构,学习使用光栅光谱仪。</p><p>(3)测量钨灯和汞灯在可见光范围的光谱。</p><p>(4)测定光栅光谱仪的色分辨能力。</p><p>(5)测定干涉滤光片的光谱透射率曲线。</p><p><strong>【实验仪器】</strong></p><p>&nbsp;WDS-3平面光栅光谱仪(200~800nm),汞灯,钨灯&氘灯组件,干涉滤光片等。</p>

实验原理

<p><strong>1.平面反射式闪耀光栅原理</strong></p><p>(1)平面反射式光栅与光栅方程&nbsp;</p><p>假定刻槽的横断面为锯齿形,如图所示。</p><p><img alt="" height="238" src="/files/testpaper/106/2022/03-23/1549299bddc2130999.jpg" width="400" /></p><p>设光栅共有N条刻槽,光栅常数为d,光栅法平面为n,槽面法线为n&rsquo;,槽面与光栅平面的夹角为&gamma;。平行光以入射角i入射到光栅上,经光栅衍射后,考虑&theta;方向上的光的光强I<sub>&theta;</sub>。理论上可以推出 I<sub>&theta;</sub>正比于(sin&alpha;/&alpha;)&sup2;(sinN&beta;/sin&beta;)&sup2;,其中&alpha;=&pi;a[sin&theta;+sini-(cos&theta;+cosi)tan&gamma;]/&lambda;,&beta;=&pi;d(sin&theta;+sini)/&lambda;,sin&alpha;/&alpha;是单条刻槽衍射造成的,通常叫作单槽衍射因子。(sinN&beta;/sin&beta;)是各槽之间的干涉造成的,叫作槽间干涉因子。</p><p>主极大出现在sin&beta;=0的方向上,即&beta;=k&pi;,可得到d(sin&theta;+sini)=k&lambda;。</p><p>在常用的平面光栅光谱仪里,谱板中心到光栅中心的连线与入射光线在同一平面内,因此,对于所获取的谱线来说,衍射角&theta;时间上都可以当作等于入射角i。在这种情况下,2dsin&theta;=k&lambda;,当k&lambda;值相同的谱线,衍射角度&theta;相同。</p><p>(2)闪耀问题</p><p>光栅相对光强分布曲线如图。</p><p><img alt="" height="214" src="/files/testpaper/106/2022/03-23/1549520d7661388998.jpg" width="400" /></p><p>由光强分布公式可见,当&alpha;=0时,干涉因子的值最大, 这就是说光强I<sub>&theta;</sub>极大值出现在&alpha;=0的方向上,这样可以推出光强最大的方向就是槽面反射定律所规定的方向。干涉图像的位置不受反射面形状的控制,只由各个面对应点的光程差决定,因此,0级光谱出现在光栅平面反射定律所规定的方向上,而闪耀方向则是刻槽平面的反射方向。在常用的平面光栅摄谱仪里,所拍摄的光谱满足&theta;=i,可以推出这时有&theta;=i=&gamma;,&lambda;=2dsin&gamma;/k,这个波长叫作闪耀波长。能使闪耀方向落在0级光谱以外的光栅叫作闪耀光栅。</p><p>选择光栅主要考虑如下因素:</p><p>1)闪耀波长:闪耀波长为最大衍射效率点,因此,选择光栅时应尽量选择闪耀波长在实验所需要波长附近。如实验为可见光范围,可选择闪耀波长为500nm。</p><p>2)光栅刻线:光栅刻线多少直接关系到光栅分辨率,刻线多光谱分辨率高,刻线少光谱覆盖范围宽,两者要根据实验灵活选择。</p><p>3)光栅效率:光栅效率是衍射到给定级次的单色光与入射单色光光强的比值。光栅效率越高,信号损失越小。为提高此效率,除提高光栅制作工艺外,还采用特殊镀膜来提高反射效率。</p><p><strong>2.&nbsp;平面光栅光谱仪结构与组成</strong></p><p>(1)光学系统</p><p>&nbsp;光源发出的光束进入入射狭缝S1,S1位于反射式准光镜M1的焦面上,通过S1射入的光束经M1反射成平行光束投向平面光栅G上,衍射后的平行光束经物镜M2成像在S2上。光栅G放置在一个平台上,由步进电动机带动,可以绕通过光栅平面的铅垂轴转动,从而可以改变平行光束相对于光栅平面的入射角。</p><p>(2)电子系统 电子系统主要由电源系统,光接收系统,步进电动机系统等部分组成。光接收系统由光电倍增管及放大电路构成。</p><p>1)光电倍增管的基本结构 :</p><p>光电倍增管是利用光电阴极K与多级&ldquo;次级发射极&rdquo;制成的光电探测器。光电倍增管的工作原理图如图。</p><p><img alt="" height="241" src="/files/testpaper/106/2022/03-23/1550168970df894591.jpg" width="400" /></p><p>光电阴极K接负高压,阳极A接电源正极,各倍增极电压由直流电源经分压电阻供给,每相邻增极间的工作电压控制在70~100V左右,负载R<sub>L</sub>接在阳极处。当光照射到光电阴极K上时,只要光子能量大于光电阴极材料的逸出功,光电阴极表面就发射出电子,在真空中被K和D1间电场加速,而射到第一个&ldquo;初级发射极&rdquo;D<sub>1</sub>上。快速的电子轰击使D<sub>1</sub>产生二次电子发射,出射的电子数位入射电子数M倍,这些电子又在D<sub>1</sub>和D<sub>2</sub>间的电场作用下飞向D<sub>2</sub>,因而从D<sub>2</sub>上又产生新的二次电子流,它的数量是原来光电子的M&sup2;倍。如果电子逐一地在各个二次发射极下被倍增,从最后一个二次发射极D<sub>n</sub>出射的电子数将达到由K极出射的电子书的M的n次方倍,成为阳极电流,M的n次方称为光电倍增管的放大倍数。光电倍增管阳极电流在一定范围内与入射光功率成线性关系,但当入射光功率大到一定程度时,阳极电流先出现饱和现象,而后反而下降。</p><p>使用光电倍增管时,切勿使入射光太强。</p><p>2)光电倍增管的基本特性:</p><p>1.&nbsp;灵敏度:光电倍增管接收到单位辐射功率后所产生的电信号大小称为响应率,单位A/lm</p><p>2.&nbsp;暗电流:光电倍增管接上工作电压之后,在没有光照的情况下,它仍会有一个很小的阳极电流输出,此电流被称为暗电流。产生的主要原因是由光电阴极和次级的热电子发射所产生的热电流和光电倍增各极间的漏电流。</p><p>3.&nbsp;光谱特性:当光照射光电倍增管时,产生的光信号的强度与入射光的辐射功率的大小有关,也与光的波长有关。当光波长一定时,光电流的强度与辐射功率成正比。当入射光为单色光,且各单色光辐射功率一定时,光电流随波长的变化而变化,这种现象称为光电倍增管的光电特性。</p><p>(3)光栅光谱仪操作:</p><p>&nbsp;供给高压可调范围0~1000V,由光栅光谱仪主机前面板上的&ldquo;高压调节&rdquo;旋钮来控制,调节负高压、入射或出射缝的宽度,均可以改变光电倍增管所接收到的信号的大小。除此之外,其他控制和数据处理由计算机完成。</p><p>主要功能:仪器系统复位、光谱扫描、各种动作控制、测量参数设置、光谱采集、光谱数据文件管理、光谱数据的各种计算。</p><p><strong>3.色分辨率</strong></p><p>&Delta;&lambda;=b/a*&delta;&lambda;,b为峰的半宽度,a为&lambda;<sub>1</sub>、&lambda;<sub>2</sub>峰间的间隔,&delta;&lambda;=&lambda;<sub>2</sub>-&lambda;<sub>1</sub></p><p>光强I与波长&lambda;的关系曲线如图</p><p><img alt="" height="290" src="/files/testpaper/106/2022/03-23/1550433b0e77782860.jpg" width="400" /></p><p><strong>4.&nbsp;滤光片光谱特性</strong></p><p>滤光片对不同波长的光的透射能力不一样。当波长为&lambda;、光强I<sub>0</sub>(&lambda;)的单色光垂直入射到滤光片上时,透过的光强若为I<sub>T</sub>(&lambda;),我们定义其光谱透射率为T(&lambda;)=I<sub>T</sub>(&lambda;)/I<sub>0</sub>(&lambda;),若以白光为光源,经单色仪后而出射的单色光由光电倍增管接收并转化为电流,相应的光强值由测光仪显示,出射的单色光所产生的光电流i<sub>0</sub>(&lambda;)与入射光光强I<sub>0</sub>(&lambda;)、单色光的光谱透射率T<sub>0</sub>(&lambda;)和光电器件的光谱响应率S(&lambda;)成正比。</p><p>i<sub>0</sub>(&lambda;)=KI<sub>0</sub>(&lambda;)T<sub>0</sub>(&lambda;)S(&lambda;),K为比例系数,现将光谱透射率为T(&lambda;)的滤光片插入光路,放置在入射狭缝之间,光电流变为i&gamma;(&lambda;)=KI0(&lambda;)T(&lambda;)T0(&lambda;)S(&lambda;),可得T(&lambda;)=I&gamma;(&lambda;)/I<sub>0</sub>(&lambda;)。透射率曲线通带半宽度越窄,说明滤光片的单色性越好,中心波长&lambda;<sub>0</sub>、通带半宽度&Delta;&lambda;以及峰值透射率T<sub>0</sub>为滤光片的三个特征量。</p><p>透射率T(&lambda;)随波长变化的曲线如图</p><p><img alt="" height="292" src="/files/testpaper/106/2022/03-23/1550582ab3d7447979.jpg" width="400" /></p>

实验步骤

<p>实验内容为测量汞灯和钨灯光谱,了解两种光源的光谱特性;测量光栅光谱仪的色分辨率;测量滤光片的光谱特性。</p><p><strong>1. 准备工作:</strong></p><p>1) 开机前,先缓慢旋转入射狭缝宽度调节旋钮,用眼睛直接观察入射狭缝宽度的示值与狭缝宽度之间的对应关系,以免调节时搞反方向而将狭缝关死。打开光栅光谱仪电源,高压调节到-300~-600V(不要小于-400V)之间,入射缝、出射缝缝宽均预置为0.15 ~0.30mm 之间,打开氘灯,打开计算机,打开程序。</p><p>在计算机界面上双击快捷方式&ldquo;WDS系列组合式多功能光栅光谱仪&rdquo;,根据提示进行系统复位操作。复位后按&ldquo;确定&rdquo;进入操作主界面。</p><p>2) 测量参数设置:在&ldquo;文件&rdquo;菜单中,或通过工具栏的快捷按钮进入&ldquo;参数设置&rdquo;界面。测量模式:选择能量模式,示值范围在0.0~4095.0,测量中如果信号超出此范围,需要适当调整缝宽和高压值。</p><p>扫描速度:扫描速度分为不同的档,例如&ldquo;最快&rdquo;档时,仪器会每隔1nm记录一个数据。实验时需根据测量需求进行选择,如在需要初步了解全部谱线特征时可选择&ldquo;快速&rdquo;或者&ldquo;中速&rdquo;,需要精细测量时可选择&ldquo;慢速&rdquo;或者&ldquo;很慢&rdquo;。</p><p>扫描方式:选择重复扫描1次</p><p>波长范围:最大范围200~800nm ,可根据实验需要设定上下限。</p><p>光谱带宽:系统设置为手动,即根据需要对出射,入射狭缝宽度进行相应的设置。</p><p>增益:选择&ldquo;1&rdquo;。</p><p><strong>2. 校准光谱仪的波长指示值</strong></p><p>利用氘灯波长值为486. 0nm的谱线校准光谱仪。光源采用氘灯,选择快速扫描,在&ldquo;测量方式&rdquo; 菜单中,或工具栏的快捷按钮中点选&ldquo;光谱扫描&rdquo;进行快速全谱扫描。点选后,光谱仪开始自动扫描。应根据能量信号大小手工调节负高压值、入射狭缝宽度和出射狭缝宽度,获得适当大小的能量信号,然后再选择中速扫描氘灯光谱。如果在扫描过程中进行中断及修改参数等操作,再次扫描时系统仍然继续从上一次的波长扫起。</p><p>&ldquo;数据处理&rdquo;菜单中包括&ldquo;读取数据&rdquo;&ldquo;峰值检索&rdquo;&ldquo;刻度扩展&rdquo;。&ldquo;读取数据&rdquo;时,可以采用列表方式或光标方式读取当前图谱的横、纵坐标。&ldquo;列表方式&rdquo;会在屏幕上给出包含波长和强度值的数据表格,&ldquo;光标&rdquo;读取时,将光标置于谱线上的某一个数据点处,相应的波长和强度值会在界面下方显示。&ldquo;峰值检索&rdquo;可以根据输人的峰值高度,自动检索出当前图谱文件中在一定范围内的峰值,并给出结果。&ldquo;刻度扩展&rdquo;可以对当前横纵坐标的起始、终止刻度在系统允许的范围内进行相应的放大或缩小。</p><p>利用&ldquo;数据处理&rdquo;菜单的功能读出测量的氘灯光谱谱线波长。如果光谱仪给出的尔灯谱线波长值与标准波长有偏差,用&ldquo; 系统操作&rdquo;菜单中的&ldquo; 波长校正&rdquo;功能进行校正,即在&ldquo;当前波长偏差值&rdquo;对话框中输人仪器显示的波长值与标准波长值的差值,点击确定即可。</p><p>校准后,重新扫描氘灯光谱,并保存为TXT 文件。</p><p><strong>3. 汞灯光谱和光谱仪分辨率的测量</strong></p><p>1)入射缝宽和出射缝宽设定在0.15 -0. 20mm之间,负高压-300~ -600V之间。</p><p>2)移去钨灯&amp;氘灯组件,将汞灯置于入射狭缝前,进行快速全谱扫描.根据光谱测量结果进一步调节狭缝宽度、负高压等参数,使得记录的谱线高度适当,再进行一次慢速全谱扫描,保存实验数据为TXT文件。</p><p><strong>4. 滤色片光谱特性的测量</strong></p><p>移去汞灯,将钨灯置于入射狭缝前,选择&ldquo;最快&rdquo;档进行全谱扫描,根据扫描续用判断是否需要重新设定缝宽、负高压等参数,以得到合适的图谱,保存实验数据为TXT文件。然后保持仪器各参数不变,加入滤光片,再扫描一次图谱,保存实验数据为TXT文件。最后,在无光照射下扫描一次图谱(暗电流值),保存实验数据为TXT文件。</p><p><strong>5. 退出系统与关机</strong></p><p>测试结束时,将负高压调节至零。单击菜单栏中&ldquo;文件/退出系统&rdquo;,按照提示关闭电源退出仪器操作系统。</p>

5.10 光电效应

实验目的和仪器

<p><strong>【实验目的】</strong></p><p>1)&nbsp; 观察光电效应现象,了解光电效应的基本实验规律,加深对光的量子性的理解。</p><p>2)&nbsp; 测量普朗克常数 h,逸出功和红线频率。</p><p><strong>【实验仪器】</strong></p><p>ZKY-GD-4光电效应实验仪</p>

实验原理

<p>研究光电效应的实验装置中的真空光电管示意图如图所示:</p><p><img alt="" height="171" src="/files/testpaper/106/2022/03-29/183401930fda556310.jpg" width="400" /></p><p>具有适当频率的入射光通过窗口W照射到光电管金属阴极K上,发射出光电子。这些光电子在电场作用下向金属阳极A迁移形成光电流。改变外加电压U<sub>AK</sub>(也称光电管电压),测量出光电流 I 的大小,就可得到光电管的伏安特性曲线。</p><p><strong>(1)光电管电压对光电流的影响</strong></p><p>施加在阳极和阴极之间的电压对光电流大小的影响如图所示:</p><p><img alt="" height="239" src="/files/testpaper/106/2022/03-29/183422ed846f145533.jpg" width="400" /></p><p>对于给定的入射光频率和光强,光电流的大小随施加在阳极与阴极之间的电压的增大而增大,当所有光电子达到阳极A,光电流达到最大值,这个电流被称为</p><p>饱和电流。当电压减小时,光电流减小,但减小至零时,光电流不为零。这表明,即使没有加速电压,部分光电子依靠自己的能力也能达到阳极A。当在阳极</p><p>与阴极间施加一个负电压时,在某一特定电压U<sub>s</sub>,光电流为零,这个电压被称为截止电压。对于光强不同的入射光,截止电压相同,表明入射光的光强对截止电压没有影响。</p><p><strong>(2)入射光强对光电流的影响</strong></p><p>对于给定的入射光频率,饱和光电流I<sub>M</sub>的大小随入射光强P的增大而线性增大。如图所示:</p><p><img alt="" height="300" src="/files/testpaper/106/2022/03-29/1834499ea92a641840.jpg" width="400" /></p><p><strong>(3)入射光频率对光电流的影响</strong></p><p>对于给定的入射光强,饱和光电流不依赖于入射光的频率。由于光电流的大小只依赖于逸出的光电子的数量,因而只依赖于入射的光子的数量,而与光子的</p><p>能量无关,如图所示:</p><p><img alt="" height="318" src="/files/testpaper/106/2022/03-29/183506a42ef8502920.jpg" width="400" /></p><p><strong>(4)入射光频率对截止电压的影响</strong></p><p>对于给定的入射光强,光电流随施加在阳极与阴极之间的电压的增大而增大,直至饱和。但对于不同频率的入射光,饱和电流相同。随着光电电压减小,光</p><p>电流减小,并在截止电压时减小为零,不同频率的入射光,截止电压不同。入射光频率越高,截止电压越高。</p><p><strong>(5)截止频率</strong></p><p>截止电压U<sub>s</sub>与频率&nu;如图所示:</p><p><img alt="" height="322" src="/files/testpaper/106/2022/03-29/1835291a5f00668679.jpg" width="400" /></p><p>两者成正比关系,但直线并不通过原点。这表明有一个最小频率&nu;0存在,当入射光频率低于该值时,不论光的强度如何大,都没有光电流产生。这个最小频</p><p>率被称为截止频率,也称为红限,&nu;<sub>0</sub>随不同金属而异。</p><p><strong>(6)瞬时过程</strong></p><p>光电子发射过程是瞬时过程。即具有适当频率的光一照射到金属上,立即就有光电子发射出来,所经过的时间只有10<sup>-8</sup>s量级。</p><p><strong>(7)一对一发射</strong></p><p>光电子的发射是一对一发射,即每一个具有适当频率的光子发射一个光电子。</p><p>以上就是光电效应的基本实验事实,为了解释光电效应现象,爱因斯坦提出了光电效应方程。按照爱因斯坦的光量子理论,光能并不像电磁波理论所想象的</p><p>那样,分布在波振面上,而是集中在被称为光子的微粒上,但这种微粒仍然保持着频率(或波长)的概念,频率为&nu;的光子具有的能量为E=h&nu;,h为普朗克常</p><p>量。当具有能量h&nu;的光子照射到金属表面上时,光子的能量一次被金属中的电子全部吸收,而无需积累能量的时间。电子把这一能量的一部分用来克</p><p>服金属表面对它的吸引力(即表面势垒)而脱离金属表面,这一部分能量也叫做逸出功(A=h&nu;),余下的能量就变成电子离开金属表面后的动能。按照能量</p><p>守恒原理,爱因斯坦提出了著名的光电效应方程:h&nu;=1/2*m*v<sub>0</sub><sup>2</sup>+A&nbsp; 式中,1/2*m*v<sub>0</sub><sup>2</sup>为光电子获得的初始动能。考虑到逸出功A=h&nu;<sub>0</sub>,将等式改写:</p><p>1/2*m*v<sub>0</sub><sup>2</sup>=h(&nu;-&nu;<sub>0</sub>) 。实验中可根据爱因斯坦光电效应方程来检验光电效应的实验规律</p><p>1)如果&nu;&lt;&nu;<sub>0</sub>,则 1/2*m*v<sub>0</sub><sup>2&nbsp;</sup>为负值,这是不可能的。所以,只有当&nu;&gt;&nu;<sub>0</sub>时,才会有光电子发射。</p><p>2)由于一个光子发射一个电子,所以每秒钟发射的光电子数正比于入射光的强度。</p><p>3)由于h和&nu;0为常数,1/2*m*v<sub>0</sub><sup>2&nbsp;</sup>正比于&nu;,这表明光电子的动能直接正比于入射光的频率。</p><p>4)光电子的发射是由于光子和电子的碰撞,光子的入射和光电子的发射之间不可能有明显的时间滞后,故该过程是瞬时过程。</p><p>由上述讨论可见,爱因斯坦的光量子理论成功地解释了光电效应现象实验规律。</p><p>入射到金属表面的光频率越高,发射出的电子动能越大,所以即使阳极电位比阴极电位低时也会有电子到达阳极形成电流,直至阳极电位低于截止电压,光</p><p>电流才为零,此时有如下关系:e|U<sub>s</sub>|=1/2*m*v<sub>0</sub><sup>2</sup>。</p><p>推得:|U<sub>s</sub>|=h&nu;/e-A/e。这表明截止电压|U<sub>s</sub>|是光波频率的线性函数,直线斜率k=h/e,截距为A/e。只要用实验方法得出不同的光波频率对应的截止电压,求出直线斜率和截距,即可求得普朗克常量h,逸出功A和红限频率&nu;<sub>0</sub></p>

实验步骤

<p><strong>测试前的准备:</strong></p><p>1)将汞灯及光电管暗箱遮光盖盖上,再将光电效应实验仪及汞灯的电源接通,预热20min。</p><p>2)调整光电管与汞灯距离约为40cm并保持不变。</p><p>3)用专用连接线将光电管暗箱电压输入端与实验仪电压输出端(后面板上)连接起来(红&mdash;红,蓝&mdash;蓝)</p><p>4)使&ldquo;手动/自动&rdquo;模式键处于手动模式。将光电效应实验面板上&ldquo;电流量程&rdquo;选择开关置于所选档位(如&ldquo;10<sup>-13</sup>A&rdquo;档、&ldquo;10<sup>-10</sup>A&quot;档等),仪器充分预热后,进行测试前调零。实验仪在开机或改变电流量程后, 会自动进入调零状态。调零时应将光电管暗箱电流输出端K与实验仪微电流输入端(后面板上)断开,旋转&ldquo;调零&rdquo;旋钮使电流指示为000.0。</p><p>5)用高频匹配电缆将光电管暗箱电流输出端K与实验仪微电流输入端连接起来,按实验仪面板上的&ldquo;调零确认/系统清零&rdquo;键,系统进入测试状态。</p><p><strong>(1)测量光电管在不同波长光照射下的伏安特性&nbsp;</strong></p><p>测量波长为 365. 0mm, 404.7mm,435.8nm,546.1nm.577. 0nm的光照射下光电管的伏安特性曲线。研究入射光频率对光电流的影响。</p><p><strong>具体实验步骤如下:&nbsp; </strong></p><p>1)光电效应实验仪面板上</p><p>&ldquo;伏安特性测试/截止电压测试&rdquo;状态健应为伏安转性测状态。&ldquo;电流量程&rdquo;开关选择&ldquo;10<sup>-10</sup>&rdquo;档,调零。此时电压表显示U<sub>AK</sub>的值,单位为伏;电流表显示对应的光电流值,单位为所选的&ldquo;电流量程&quot;单位。用电压调节键&ldquo;&larr;&rdquo;、&ldquo;&uarr;&rdquo;、&ldquo;&rarr;&rdquo;、&ldquo;&darr;&rdquo;可以调节U<sub>AK</sub>的值,其中&ldquo;&larr;&rdquo;、&ldquo;&rarr;&rdquo;键用于选择调节位,&ldquo;&uarr;&rdquo;、&ldquo;&darr;&rdquo;键用来调节对应位的数值的大小。</p><p>2)将直径4mm的光阑及365nm的滤色片装在光电管暗箱光输入口上。打开汞灯遮g盖。电压测量范围为-1 -50V,从低到高调节电压,合理分布测量点,以给出完整的光电管电压-光电流变化曲线以及关键特征,将实验数据记录到表中。</p><p>3) 依次换上 405nm、436nm ,546nm ,577mm 的滤色片,重复第 2 )步的测量。</p><p>4)测量截止电压</p><p>理论上,测出不同频率的光照射下,阴极电流为零时对应的U<sup>AK</sup>,其绝对值即是该频率的截止电压,然而一般情况下由于光电管的阳极反向电流、暗电流、本底电流及极间接触电位差的影响,所测电流并非阴极电流,因而电流为零时对应的U<sub>AK</sub>也并非截止电压。</p><p>&nbsp;光电管制作过程中阳极往往被污染, 活上少许阴极材料,入射光照射阳极或入射光从阴极反射到阳极后都会造成阳极光电子发射,U<sub>AK </sub>为负值时,阳极发射的电子向阴极迁移构成了了阳极反向电流</p><p>&nbsp;暗电流和本底电流是热激发产生的光电流与杂散光照射光电管产生的光电流,可以在光电管制作,或测量过程中采取适当的措施以减小或消除它们的影响。</p><p>&nbsp;极间接触电位差与入射光频率无关,只影响U<sub>s</sub>的准确性,不影响U<sub>s</sub>, -v直线的斜率,对测定h无影响。</p><p>此外,由于截止电压是光电流为零时对应的电压,若电流放大器灵敏度不够,或稳定性不好,也会给测量带来较大误差。</p><p>本实验所用仪器采用新型结构的光电管。其特殊结构使光不能直接照射到阳极,由阴极反射照到阳极的光也很少,再加上采用新型的阴、阳极材料及制造工艺,使得阳极反向电流大大降低,暗电流和本底电流都很小。因此,本实验仪器在测量不同频率光波的截止电压U<sub>s</sub>时,可以采用&ldquo;零电流法&rdquo;或&ldquo;补偿法&rdquo;。&ldquo;零电流法&rdquo;是直接将各频率光照射下测得的电流为零时对应的电压UAK的绝对值作为截止电压U<sub>s</sub>。&ldquo;补偿法&rdquo;是调节电压U<sub>AK</sub>使电流为零后,保持U<sub>AK</sub>不变,遮挡汞灯光源,此时测得的电流I<sub>1</sub>为电压接近截止电压时的暗电流和本底电流。重新让汞灯照射光电管,调节电压U<sub>AK</sub>使电流值至I<sub>1</sub>,将此时对应的电压U<sub>AK</sub>的绝对值作为截止电压U<sub>s</sub>。此法可补偿暗电流和本底电流对测量结果的影响。</p><p>这里将采用零电流法测量波长分别为365. 0nm、404. 7nm、435. 8nm、546. 1nm,577. 0nm入射光的截止电压U<sub>s</sub>。具体测量步骤如下:</p><p>①光电效应实验仪面板上&ldquo;伏安特性测试/截止电压测试&rdquo;状态键切换为截止电压测试状态,并按&ldquo;调零确认系统清零&rdquo;键进行确认。&ldquo;电流量程&rdquo;开关应处于&ldquo;10<sup>-13</sup>A&rdquo; 档</p><p>②将直径4mm的光阑及365nm的滤色片装在光电管暗箱光输入口上,打开汞灯道光盖。从低到高调节电压(绝对值减小),观察电流值的变化,寻找电流为零时对应的UAK值,以其绝对值作为该波长对应的的U<sub>s</sub>值,将数据记于表中。</p><p>③依次换上405nm,436nm ,546nm 577nm的滤色片,重复第2步测量。</p><p><strong>(2) 入射光光强P对饱和光电流I<sub>M</sub>的影响&nbsp;&nbsp;</strong></p><p>采用改变光阑直径的办法改变照射到光电管入射窗口的光强。入射光波长为435.8nm。光电效应实验仪面板上&ldquo;伏安特性测试/截止电压测试&rdquo;状态键为伏安特性测试状态。&ldquo;电流量程&rdquo;开关选择&ldquo;10<sup>-10</sup>A&rdquo;档,重新调零。在U<sub>AK</sub>为50V时,测量并记录光阑直径分别为2mm,4mm,8mm时对应的电流值,并记录于表中。</p>

5.11 电子电荷e值的测定

实验目的和仪器

<p><strong>实验目的:</strong></p><p>(1)学习密立根油滴实验的设计思想.</p><p>(2)测量电子的电荷量,验证电荷的不连续性。</p><p><strong>实验仪器:</strong></p><p>CCD显微密立根油滴实验仪、钟表油、油喷雾器等。</p>

实验原理

<p>密立根油滴实验测定电子电荷的基本设计思想是使带电油滴在测量范围内处于受力平衡的状态,通过对带电微小油滴的受力分析,把对微小油滴所带电荷的测量转化为对油滴运动速度的测量。</p><p>油滴实验原理图如图:</p><p><img alt="" height="186" src="/files/testpaper/106/2022/04-07/22540809664a159091.jpg" width="400" /></p><div>
<div id="studentAnswer3923">
<p>A、B为两块相距为d的平行板,A板中央有一小孔,能让微小的油滴从孔中落入到极板间。A、B板如不加电压,油滴将因重力作用而自由下落,当A、B两板加上电压且油滴又带有电荷(由喷雾器喷出油滴时因摩擦可使之带电)时,则油滴要受到电场力的作用,改变电场的方向可使油滴向上或向下运动。由于空气的黏滞性,油滴运动一小段距离后就要做匀速运动。油滴进入匀速运动前的变速运动时间非常短,小于0.01s,与计时器精度相当,因此在测量上可以认为油滴从静止开始运动立刻进入匀速运动。</p><p>设油滴质量为m,电荷量为q,当A、B两板不加电压,并用开关 S 将其短路使二极板无电场时,则在重力作用下,油滴下落,但空气的黏滞性对油滴所产生的阻力与速度成正比,油滴走一小段距离速度达到一定值v<sub>g</sub>时。粘滞力F就与重力平衡,即F=mg,根据斯托克斯定律:F=6&pi;&eta;rv<sub>g</sub>。式中,&eta;为空气的黏度;r为油滴半径。这时6&pi;&eta;rv<sub>g</sub>=mg。如果在平行极板上加电场E,设电场力qE与重力方向相反,油滴受电场力作用加速上升,由于空气阻力作用,上升一段距离后,油滴所受的空气阻力、电场力与重力达到平衡(空气浮力忽略不计),油滴将以匀速上升,速率为v<sub>e</sub>,有&nbsp; &nbsp;6&pi;&eta;rv<sub>e</sub>=qE-mg。</p><p>可得&nbsp; &nbsp;q=mg/E *( (&nbsp;v<sub>g</sub>+v<sub>e</sub>)/v<sub>g</sub></p><p>由于空气巾悬浮和表面张力作用,可将油滴视作圆球,其质量为&nbsp; &nbsp; m=4/3*(&pi;r<sup>3</sup>)(<span class="mq-math-mode" latex-data="\rho"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><var mathquill-command-id="3">&rho;</var></span></span><span>&nbsp;-</span><span class="mq-math-mode" latex-data="\sigma"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><var mathquill-command-id="5">&sigma;</var></span></span><span>&nbsp;</span>)</p><p>式中,&rho;为油的密度; &sigma;为空气密度。</p><p>由喷雾器喷出的小油滴的半径r很小,为微米数量级,难于直接测量,故采用间接测量的办法,利用斯托克斯定律,可得:</p><p>&nbsp;r=((9/2)&eta;v<sub>g</sub>/((<span class="mq-math-mode" latex-data="\rho" style="font-size: 18.4px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 12.8375px;"><var mathquill-command-id="3">&rho;</var></span></span>&nbsp;-<span class="mq-math-mode" latex-data="\sigma" style="font-size: 18.4px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 13.075px;"><var mathquill-command-id="5">&sigma;</var></span></span>&nbsp;)g))<sup>1/2</sup></p><p>考虑到油滴非常小,空气已不能看作是连续介质,空气的黏度应修正为</p><p>&eta;&#39;=&eta;/(1+b/(pr))</p><p>b为修正常数,p为空气压强。</p><p>实验时取油滴匀速下降和匀速上升的距离相等,设为l,测出油滴匀速下降的时间t<sub>g</sub>和匀速上升的时间t<sub>e</sub>,则:</p><p>v<sub>g</sub>=l/t<sub>g&nbsp; ,</sub>v<sub>e</sub>=l/t<sub>e</sub></p><p>将以上各式以及E=U/d(U为A,B两极板上所加电压、d为板板间距)&nbsp; 代入式中可得 :</p><p><img alt="" height="59" src="/files/testpaper/106/2019/05-13/012606e845e2839770.jpg" width="250" /></p><p>由该式可算出油滴所带电荷,式中油滴半径可以近似计算。(即不对黏度进行修正)。</p><p>该式为动态(非平衡)法测油滴所带电荷的计算公式。还可以用静态(平衡)法测量油滴所带电荷。</p><p>用静态法测量油滴电荷时,需要调整两极板上所加的电压,使油滴处于静止状态。此时v<sub>e</sub>=0,即。t<sub>e</sub>逼近&infin;,可得</p><p><img alt="" height="74" src="/files/testpaper/106/2019/05-13/012629536650764736.jpg" width="250" /></p><p>为了求电子电荷e,对实验测得的各个油滴所带电荷q求最大公约数,最大公约数就是电子电荷e的值。也可以测同油滴所带电荷的改变量△q(可以用紫外线或放射源照射油滴,使它所带电荷改变),这时△q应近似为某一最小单位的整数倍,此最小单位即为元电荷e。</p>

实验步骤

<p>学习控制油滴在视场中的运动,并选择合适的油滴测量元电荷。要求至少测量5个不同的油滴,每个油滴测量5次。</p><p><strong>1.仪器调节</strong></p><p><strong>(1)&nbsp;水平调整</strong></p><p>调整实验仪主机上的调平螺钉旋钮(俯视时,顺时针旋转平台降低,逆时针旋转平台升高),直到水准泡正好处于中心(注意:不要旋转水准泡上的旋钮),即通过水准仪将实验平台调平,使平衡电场方向与重力方向平行以免引起实验误差。极板平面是否水平决定了油滴在下落或提升过程中是否发生前后、左右的漂移。</p><p><strong>(2) 喷雾器调整&nbsp;&nbsp;</strong>&nbsp;</p><p>将少量钟表油缓慢的倒入喷雾器的储油腔内,使钟表油湮没提油管下方。油不要太多,以免实验过程中不慎将油倾倒至油滴盒内堵塞落油孔。将喷雾器竖起,用手挤压气囊,使得提油管内充满钟表油。</p><p><strong>(3) 仪器硬件接口连接&nbsp;&nbsp;</strong></p><p>主机接线:电源线接交流220V/50Hz。监视器:&nbsp;Q9视频线缆端接监视器上的&ldquo;VIDE0&rdquo;&nbsp;插座,另一端接主机上的&nbsp;&ldquo;&nbsp;视屏输出&rdquo;端口。监视器上前面板调整旋钮自左至右依次为显示开关、返回键、方向键、菜单键(建议亮度调整为20,对比度调整为100)。</p><p><strong>(4) 实验仪联机使用</strong></p><p>1) 打开实验仪电源及监视器电源,监视器出现仪器名称界面。</p><p>2) 按主机上任意键,监视器出现参数设置界面。此时,按主机面板上的&ldquo;平衡/提升&rdquo;按钮,选择平衡法或动态法,然后设置重力加速度、油密度、大气压强、油滴下落距离等。主机面板上的&ldquo;&lt;-&rdquo;为左移键、&ldquo;-&gt;&rdquo;为右移键、&ldquo;+&rdquo;为数据设置键。</p><p>3) 按主机面板上的&ldquo;确认&rdquo;键,监视器上出现实验界面。</p><p>此时,主机面板上的计时&ldquo;开始/结束&rdquo;按钮的&ldquo;结束&rdquo;指示灯亮,&ldquo;0V/工作&rdquo;按钮的&ldquo;0V&rdquo;指示灯亮,&ldquo;平衡/提升&rdquo;按钮的&ldquo;平衡&rdquo;指示灯亮。</p><p><strong>(5)&nbsp;CCD&nbsp;成像系统调整&nbsp;&nbsp;&nbsp;</strong></p><p>从喷雾口喷入油雾,此时监视器上应该出现大量运动油滴的像。若没有看到油滴的像,则需调整调焦旋钮或检香喷雾器是否有油雾喷出。CCD显微镜对焦时先将显微镜筒上小黑圈外缘与防风罩边缘大致对齐,然后喷油后再稍稍前后微调即可。如果按上述方法调节始终看不到油滴,则首先要检查喷雾器是否能喷出油滴,将喷雾器对着空中喷一下,用眼睛直接观察有没有油雾即可确定。如果喷雾器喷雾没有问题,则可以打开油雾杯,先清除堵塞上电极落油孔的油污,然后在油滴盒中放一根细金属(此时要先将平衡电压放到&rdquo;0V&ldquo;档),调显微镜焦距旋钮,使金属丝的像清晰即可。焦距调好后,在使用过程中,前后调焦范围不要过大。</p><p><strong>2. 熟悉实验界面</strong><br />
在完成参数设置后,按&ldquo;确认&rdquo;键,监视器显示实验界面。平衡法和动态法的实验界面略有不同。<br />
实验界面显示的主要内容有:</p><p>1)极板电压:实际加到极板的电压,显示范围:&nbsp;&nbsp;0&nbsp;~9999V。</p><p>2)经历时间:定时开始到定时结束所经历的时间,显示范围:&nbsp;0~99.99s。<br />
3)电压保存提示: 在每次完整的实验后显示将要作为结果保存的电压。当保存实验结果后(即按下&ldquo;确认&rdquo;键)自动清零。显示范围同极板电压。<br />
4)保存结果显示: 显示每次保存的实验结果,共5次,显示格式与实验方法有关。</p><p>平衡法:(平衡电压)&nbsp; &nbsp;动态法:(提升电压)(平衡电压)</p><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (下落时间)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (上升时间)(下落时间)</p><p>当需要删除当前保存的实验结果时,按下&ldquo;确认&rdquo;键2s以上,当前结果被清除(不能连续删)。</p><p>5)下落距离设置:显示当前设置的油滴下落距离。当需要更改下落距离的时候,按住&ldquo;平衡/提升&rdquo;键2s以上,此时距离设置栏被激活(注意:动态法1步骤和2步骤之间不能更改),通过&ldquo;+&rdquo;键(即&ldquo;平衡/提升&rdquo;键)修改油滴下落距离,然后按&ldquo;确认&rdquo;键确认修改。距离标志也会相应变化。</p><p>6)距离标志: 显示当前设置的油滴下落距离,在相应的格线上做数字标记,显示范围: 0.2~1. 8mm。垂直方向视场范围为2.0mm,分为10格,每格0.2mm。</p><p>7)实验方法: 显示当前的实验方法(&nbsp;平衡法或动态法),在参数设置画面次设定。若要改变实验方法,只有重新启动仪器(关、开仪器电源)。对于平衡法,实验方法栏仅显示&ldquo;平衡法&rdquo;字样;对于动态法,实验方法栏除了显示&ldquo;动态法&rdquo;以外还显示即将开始的动态法步骤。如将要开始动态法第一步(油滴下落),实验方法栏显示&ldquo;1 动态法&rdquo;。同样,当做完动态法第步骤 ,即将开始第二步骤时,实验方法栏显示 &ldquo;2 动态法&rdquo; 。</p><p><strong>3. 选择适当的油滴并练习控制油滴</strong></p><p><strong>(1) 选择适当的油滴&nbsp;&nbsp;</strong></p><p>要做好油滴实验,所选的油滴大小要适中。大的油滴虽然明亮,但一般带的电荷多,下降或提升速度太快,不容易测准确。太小则受布朗运动的影响明显,测量时涨落较大,也不容易测准确。因此应该选择质量适中而带电不多的油滴。建议选择平衡电压在100 ~400V之间、下落时间在10s (当下落距离为1mm时)左右的油滴进行测量。</p><p>具体操作: 将&ldquo;计时&rdquo;按键置为&ldquo;结束&rdquo;,工作状态置为&ldquo;工作&rdquo;,&ldquo;平衡/提升&rdquo;键置为平衡,调节电压旋钮将电压调至400V左右,喷入油雾,调节调焦旋钮,使监视器上显示大部分油滴,可以看到带电多的油滴迅速上升出视场,不带电的油滴下落出视场,约10s后油滴减少。选择那种上升缓慢的油滴作为暂时的目标油滴,切换&ldquo;0V/工作&rdquo;键,这时极板间的电压为0V,在暂时的目标油滴中选择下落1格的时间约2s左右的油滴作为最终的目标油滴,调节调焦旋钮使该油滴最为清晰(即最小最亮)。</p><p>注意:喷油时,先在喷雾器中注人几滴油,然后将油从油雾室旁喷雾口中喷入,监视器视场中将出现大最油满,有如夜空繁星。注意喷雾器内的油不可装得大满,否则会喷出很多&ldquo;油珠&rdquo;而不是&ldquo;油雾&rdquo;,堵塞上电极的落油孔。喷油时喷雾器的探头不要伸入喷油孔内,防止大颗粒油滴堵塞落油孔。</p><p><strong>(2) 平衡电压的确认&nbsp;&nbsp;</strong></p><p>目标油滴聚焦到最小最亮后,仔细调整平衡电压旋钮使油滴平衡在某一格线上,&nbsp;等待一段时间&nbsp;(约2min左右),观察油滴是否飘离格线。若油滴始终向同一方向飘动,则需重新调整平衡电压;若其基本稳定在格线或只在格线上下做轻微的布朗运动,则可以认为其基本达到了力学平衡。这时的电压就是平衡电压。</p><p><strong>(3) 控制油滴的运动</strong>&nbsp;&nbsp;如图所示,将油滴平衡在监视器屏幕顶端的第一条格线上。</p><p><img alt="" height="239" src="/files/testpaper/106/2022/04-07/225610a3988c590144.jpg" width="400" /></p><p>将工作状态按键切换至&ldquo;0V&rdquo;,&nbsp;绿色指示灯点亮,此时上下极板同时接地,电场力为零,油滴将在重力、浮力及空气阻力的作用下做下落运动,当油滴下落到有0标记的刻度线时,立刻按下&ldquo;计时&rdquo;键,计时器开始记录油滴下落的时间;待油滴下落至有距离标志(例如1.6)的格线时,再次按下&ldquo;计时&rdquo;键,计时器停止计时(计时位置见图),油滴将停止下落。而后,&ldquo;0V/工作&rdquo;&nbsp;按键自动切换至&ldquo;工作&quot;,&ldquo;平衡/提升&rdquo;按键处于&ldquo;平衡&rdquo;,此时,可以按下&ldquo;确认&rdquo;键将此次测量数据记录到屏幕上。将&ldquo;平衡/提升&rdquo;按键切换至&ldquo;提升&rdquo;,这时极板电压将在原来平衡电压的基础上再增加约200V电压,油滴立即向上运动,待油滴到达监视器屏幕顶端时,&ldquo;平衡/提升&rdquo;按键切换至&ldquo;平衡&rdquo;,找平衡电压,进行下一次测量。&nbsp;每颗油滴共测量5次,系统会自动计算出这颗油滴所带电荷量。在测量过程中油滴如果变得模糊,可微调显微镜调焦旋钮使其聚焦。</p><p><strong>4. 正式测量</strong></p><p>可以采用平衡测量法或动态测量法(选做)</p><p><strong>平衡测量法:</strong></p><p>1)开启主机和监视器电源,进入实验界面,将&ldquo;0V/工作&rdquo;按键切换至&ldquo;工作&rdquo;,红色指示灯点亮;将&ldquo;平衡/提升&rdquo;按键置于&ldquo;平衡&rdquo;。</p><p>2)将平衡电压调整为400V左右,通过喷雾口向油滴盒内喷入油雾,此时监视器上将出现大量运动的油滴。选取适当的油滴,仔细调整平衡电压,使其平衡在监视器屏幕顶端的第一条格线上。</p><p>平衡法示意图如图所示:</p><p><img alt="" height="239" src="/files/testpaper/106/2022/04-07/225610a3988c590144.jpg" width="400" /><br />
3) 将&ldquo;0V/工作&rdquo;按键切换至&quot;0 V&rdquo;,此时油滴开始下落,当油滴下落到有&ldquo;0&rdquo;标记的格线时,立即按下&ldquo;计时&rdquo;键,让时器启动,开始记录油满的下落时间。</p><p>4)&nbsp;当油滴下落至有距离标记的格线时(例如1.6),立即按下&ldquo;计时&rdquo;键,计时器停止计时,油滴停止移动。而后&quot;0V/工作&rdquo;按键将自动切换至&ldquo;工作&rdquo;,此时按下&ldquo;确认&rdquo;按键,这次测量的平衡电压和匀速下落时间将同时记录在监视器屏幕上。</p><p>5)将&ldquo;平衡/提升&rdquo;按键置于&ldquo;提升&rdquo;,油滴将被提升向上运动,当回到高于有&ldquo;0&rdquo;标记格线时,将&ldquo;平衡/提升&rdquo;按键置回&ldquo;平衡&rdquo;,油滴停止上升,然后重新调整平衡电压,使其静止,得到新的平衡电压</p><p>(注意:如果此处的平衡电压发生了突变,则该油滴得到或失去了电子。这次的测量无效,需从步骤②开始重新找油滴)。</p><p>6) 重复3)、4)、5),并将平衡电压及下落时间数据记录到屏幕上。当5次测量完成后,按&ldquo;确认&rdquo;键,系统将计算5次测量的平均平衡电压和平均下落时间,进而自动计算和显示出该油滴的电荷量。</p><p>7) 重复2)、3)、4)、5)、6)步,测量5颗油滴,获得每颗油滴的电荷量。<einfo></einfo></p>

5.12 弗兰克-赫兹实验

实验目的和仪器

<p><strong>【实验目的】</strong></p><p>1)观察弗兰克-赫兹管板极电流与栅级电压的关系,了解电子与原子碰撞和能量交换的过程,验证原子能级的量子化。</p><p>2)测量氩原子的第一激发电位,加深对原子能级的理解</p><p><strong>【实验仪器】</strong></p><p>FD-FH-B弗兰克-赫兹实验仪</p>

实验原理

<p>波尔提出的量子理论指出:</p><p>1)原子只能较长久地停留在一些稳定状态(简称定态)。原子处在这些状态时,不发射或吸收能量。各定态有一定的能量,其数值是彼此分立的。原子的能量不论通过什么方式改变,它只能使原子由一个定态跃迁到另一个定态。</p><p>2)原子从一个定态跃迁到另一个定态而发射或吸收辐射时,辐射频率是一定的。辐射频率&nu;由下式确定:</p><p>&nbsp; &nbsp; &nbsp; h&nu;=E<sub>m</sub>-E<sub>n</sub>(其中h为普朗克常量,其值为6.6260&times;10<sup>-34&nbsp;&nbsp;</sup>J&middot;s)</p><p>&nbsp; &nbsp; 为了使原子从低能级向高能级跃迁,可以通过具有一定频率 &nu; 的光子来实现,也可以通过具有一定能量的电子与原子非弹性碰撞,进行能量交换的方法来实现。</p><p>&nbsp; &nbsp; 设初速度为零的电子在电势差为 U 的加速电场作用下获得 eU 的能量。在充氩的弗兰克-赫兹管中,具有一定能量的电子将与氩原发生碰撞。如果以E<sub>1</sub>代表氩原子的基态能量,E<sub>2</sub>代表氩原子的第一激发态的能量,当电子与氩原子相碰撞传递给氩原子的能量恰好是&nbsp; &nbsp; eU<sub>0</sub>=E<sub>2</sub>-E<sub>1</sub></p><p>则氩原子就会从基态跃迁到第一激发态。而相应的电势差U<sub>0</sub>称为氩原子的第一激发电位。其他元素气体原子的第一激发电位也可以按此法测量得到。</p><p>&nbsp; &nbsp; 在实验中,氩原子与电子碰撞是在弗兰克-赫兹管内(F-H管)内进行的。</p><p>&nbsp; &nbsp; F-H管结构图如下:</p><p><img alt="" height="285" src="/files/testpaper/106/2022/04-17/10123204f917442059.jpg" width="400" /></p><p>&nbsp; &nbsp; 管内共设有四个电极:1. 发射电子的阴极K,它由管中的灯丝 F 通电加热而造成热电子发射; 2.用于消除空间电荷对阴极电子发射的影响以提高发射效率的第一栅极G<sub>1</sub>;3. 用于加速电子的第二栅极G<sub>2</sub>;4. 收集电子的板级P。</p><p>&nbsp; &nbsp; F-H管内空间电位分布图如下所示:</p><p><img alt="" height="220" src="/files/testpaper/106/2022/04-17/101247f801af233333.jpg" width="400" /></p><p>电子由热阴极K发出,阴极K和栅极G<sub>2</sub>之间的可调加速电压U<sub>G2K</sub>使电子加速。在板级P和栅极G<sub>2</sub>之间加有反向拒斥电压U<sub>G2P</sub>。 当电子通过栅极G<sub>2</sub>进入G<sub>2</sub>P空间时,如果能量大于e(U<sub>G2K</sub>-U<sub>G2P</sub>),就能冲过G<sub>2</sub>P空间到达板级,形成板级电流I<sub>P</sub>,为微电流计检出。如果电子在G<sub>1</sub>G<sub>2</sub>空间与氩原子发生了碰撞,电子本身剩余的能量小于e(U<sub>G2K</sub>-U<sub>G2P</sub>),则电子不能到达板级。</p><p>&nbsp; &nbsp; 实验时,使栅极电压U<sub>G2K</sub>逐渐增加并观察微电流计的电流指示。如果原子能级确实存在,而且基态与第一激发态之间有能量差,能观察到 I<sub>P</sub>-U<sub>G2K</sub>的关系曲线。</p><p>&nbsp; &nbsp; I<sub>P</sub>-U<sub>G2K</sub>的关系曲线如下图所示:</p><p><img alt="" height="302" src="/files/testpaper/106/2022/04-17/101257997397100450.jpg" width="400" /></p><p>该曲线反映了氩原子在KG<sub>2</sub>空间与电子进行能量交换的情况。当KG<sub>2</sub>空间电压逐渐增加时,电子在KG<sub>2</sub>空间被加速而取越来越大的能量。</p><p>在起始阶段由于电压较低,电子能量较小,电子的能量几乎不会减少(弹性碰撞),穿过栅极的电子形成的板极电流 I<sub>P</sub>将随着栅极电压U<sub>G2K</sub>的增加而增大,即图中oa段。图中oa段前的Oo段电压是 F-H 管的阴极K和栅极G<sub>2</sub>之间由于存在接触电位差而出现的。图中的接触电位差U<sub><span style="font-size: 13.3333px;">c</span></sub>是正的,它使整个曲线向右平移。如果接触电位差U<sub>c</sub>是负的,则整个曲线向左平移。</p><p>&nbsp; &nbsp; &nbsp; &nbsp;当KG<sub>2</sub>间的电压达到(U<sub><span style="font-size: 13.3333px;">o</span></sub>+U<sub><span style="font-size: 13.3333px;">c</span></sub>)时,电子在栅极G<sub>2</sub>附近与氩原子碰撞,将自己从加速电场中获得的能量交给氩原子,使其从基态激发到第一激发态,而电子由于自身能量所剩无几,即使穿过栅极也不能克服反向拒斥电场而被折回栅极,因此&nbsp;I<sub>P&nbsp;</sub>显著减小,如图ab段。随着栅极电压 U<sub>G2K</sub>的增加,电子的能量也随之增加,在与氩原子相碰撞后,一部分能量(E<sub>2</sub>-E<sub>1</sub>)交换给氩原子,还留下一部分能量足够克服反向拒斥电场而达到板级P,这时板级电流 I<sub>P&nbsp;</sub>又开始上升,即曲线中的bc段,直到KG<sub>2</sub>间的电压达到(2U<sub><span style="font-size: 13.3333px;">o</span></sub>+U<sub>c</sub>)时,电子在KG<sub>2</sub>空间会因与氩原子发生两次碰撞而失去2eU<sub>o</sub>的能量,又造成第二次板极电流下降,即图中的cd段。在加速电压较高的情况下,电子在运动过程中,将与氩原子发生多次非弹性碰撞,在I<sub>P</sub>-U<sub>G2K</sub>关系曲线上就表现为多次下降。</p><p>&nbsp; &nbsp; 也即凡是在&nbsp; U<sub>G2K</sub>=nU<sub><span style="font-size: 13.3333px;">o</span></sub>+U<sub>c</sub>(n为正整数)时</p><p>板极电流都会下降,形成规则起伏变化的曲线。而各次板级电流开始下降(或上升),即曲线上相邻峰(或谷)之间对应的阴极和栅极之间的电位差 (U<sub>G2K</sub>)<sub>n+1</sub>-(U<sub>G2K</sub>)<sub>n&nbsp;</sub>就是氩原子的第一激发电位U<sub><span style="font-size: 13.3333px;">o</span></sub>。曲线的极大极小的出现呈现明显的规律性,这是量子化能量被吸收的结果。原子只吸收特定能量而不是任意能量,这证明了氩原子能量状态的不连续性。</p>

实验步骤

<p>通过实验测定充氩F-H管的 I<sub>P</sub>-U<sub>G2K</sub>曲线,观察电子与氩原子碰撞和能量交换的过程,观察原子能量量子化的情况。</p><p><strong>1. 认识实验仪器</strong></p><p>FD-FH-B型弗兰克-赫兹实验仪采用充氩气的F-H管。</p><p>FD-FH-B型弗兰克-赫兹实验仪采用的是双栅柱面型四极式F-H管,结构如图所示:</p><p><img alt="" height="559" src="/files/testpaper/106/2022/04-17/1013106af23a118141.jpg" width="400" /></p><p>板极P为敷铝的铁皮圆筒;控制栅G<sub>1</sub>和加速栅G<sub>2</sub>分别采用铝丝烧制的螺旋线构成;阴极K为镍管,管的外壁敷有三元氧化物,管内有加热用的热子F,它是双向绞绕的钨丝,钨丝表面涂敷有氧化铝绝缘层。热子F和阴极K构成傍热式氧化物阴极,发射系数远大于直热式阴极。各电极同轴地固定在云母绝缘片,装入玻壳内,然后接到真空系统上抽空、除气和处理,最后充入惰性气体氩。各电极作用如下:</p><p>1)灯丝电压U<sub>F</sub>:灯丝温度对阴极的发射系数有很大的影响。阴极发射出来的电子的速度分布于阴极温度有关。阴极温度低,电子速度分布窄。</p><p>2)控制栅电压U<sub>G1K</sub>:用于消除电子在阴极附近的堆积效应,控制阴极发生的电子流的大小。U<sub>G1K</sub>过大时,会减小进入碰撞空间的电子流,导致板级电流下降,一般取1V左右。由于阴极的发射系数各不相同,而且G<sub>1</sub>与K的间距也可能略有差异,因此在实验中应选取最佳的U<sub>G1K</sub>值。</p><p>3)电子的加速电压U<sub>G2K</sub>:加速电压的上限是以管子不发生电离为界,不同的实验条件下,加速电压的上限有很大差异。</p><p>4)减速电压U<sub>G2P</sub>:使G<sub>2</sub>处的能量较低的电子不能达到板极。减速电压越大,板级电流越小,一般控制2~8V,最佳值则需要在实验中根据实测结果来选定。</p><p><strong>2. 实验测量</strong></p><p>&nbsp; &nbsp; &nbsp; &nbsp;1)调节U<sub>G2K</sub>至最小,扫描开关置于&ldquo;手动挡&rdquo;,打开主机电源,预热仪器数分钟。</p><p>&nbsp; &nbsp; &nbsp; &nbsp;2)调节U<sub>F</sub>、U<sub>G1K</sub>、U<sub>G2K</sub>电压至合适值,将&ldquo;测量设置与I<sub>P</sub>电流显示&rdquo;切换至0.1&mu;A档,调节U<sub>G2K</sub>旋钮逐渐增大U<sub>G2K</sub>,同时观察I<sub>P</sub>电流变化,可以看到至少出现7个峰。</p><p>&nbsp; &nbsp; &nbsp; &nbsp;3)选取合适的电流表档位,恰当布置实验点,分别由表头读取I<sub>P</sub>和U<sub>G2K</sub>值,将实验数据填入表中。</p>

7.4 超声波在物质中的传播与超声成像

实验目的和仪器

<div><p><strong>【实验目的】</strong></p><p>(1)掌握超声波的产生方法、传播规律和测试的原理;</p><p>(2)通过对固体弹性常数的测量掌握超声波在测试方面应用的特点;</p><p>(3)掌握超声成像的基本原理。</p><p><strong>【实验仪器】</strong></p><p>JDUT-2型超声波实验仪器、示波器、CSK-IB型铝试块、超声波成像专用试块、直探头、斜探头、5M&Phi;6专用直探头、钢板尺、耦合剂(水)</p>

实验原理

 <div><p><strong>1. 压电晶材料</strong></p><p>某些材料(晶体)在外力作用下变形时,其晶体内正负电荷中心相对位置发生移动,导致晶体两端表现出符号相反的束缚电荷,而且其电荷密度与压力成正比。这种由于压力而产生的电极化现象称为正压电效应。而这些具有压电效应的晶体,当其处于外电场作用下,由于库仑力作用致使晶体内部正负电荷中心相对位置发生移动,从而致使晶体产生形变。这种由于电极化现象而产生机械形变的现象称为逆压电效应。平常所述的压电效应是正压电效应和逆压电效应的统称。</p><p>物质的压电效应与其内部结构有关,如果某个晶体其内部结构(晶体结构)中无对称中心,这种晶体就具有压电效应的产生条件,否则,当晶体受到外力作用时,由于结构对称而导致在其形变过程中正负电荷中心无法分离。石英晶体是最早应用于声呐系统(换能器)使之产生超声波的压电材料,石英晶体的化学成分是SiO<sub>2</sub>,它可以看成由+4价的 Si 离子和 -2 价O离子组成。晶体内,两种离子形成有规律的六角形排列,如图所示。</p><p><img alt="" height="285" src="/files/testpaper/106/2022/04-27/14195174b1bc593306.jpg" width="600" /></p><p>其中三个硅正离子组成一个向右的正三角形,正电中心在三角形的重心处;三个氧负离子对(六个负离子)组成一个向左的正三角形,其负电中心也在这个三角形的重心处。当晶体不受力时,两个三角形重心重合,六角形单元是电中性的,而整个晶体由许多这样的六角形单元构成,因此晶体本身也呈电中性。</p><p>当晶体沿x方向或沿y方向受到外力作用时,上述六角形沿x方向或y方向压缩,使得正负电荷中心不再重合。尽管这时六角形单元仍然总体呈电中性,但是由于正负电荷中心不再重合,而产生电偶极矩 p。整个晶体中有许多这样的电偶极矩有序排列,使得晶体整体发生极化,晶体表面出现束缚电荷,发生正压电效应。当外力撤去时,晶体恢复原有形状,晶体极化随之消失。需要指出的是,石英晶体的压电效应是有方向性的,当外力沿z轴方向作用时,由于不能造成正负电荷中心的相对移动,因而不能产生压电效应。相反,将一个不受外力的石英晶体放置于电场中,在库仑力的作用下,正负电荷中心向相反的方向移动,最终导致晶体发生机械变形,即逆电压效应。</p><p>在压电晶体中还包括一类铁电性压电晶体,如钛酸钡( BaTiO<sub>3&nbsp;</sub>)、电陶瓷锆钛酸铅 Pb ( Zr<sub>x</sub>Ti<sub>1-x</sub>)O<sub>3</sub>(简称 PZT )。在室温下即使不受外力作用其正负电荷中心也不重合,而有一个自发偶极矩,在外力作用下这类晶体的自发偶极矩也可以发生变化从而产生压电效应。这类晶体多是由人工制成的陶瓷材料,而称为压电陶瓷。压电陶瓷一般都具有十分优异的机电性能,具有十分稳定的压电性能以及居里温度,广泛应用于制造超声换能器、水声换能器、电气换能器、陶瓷滤波器、陶瓷变压器、陶瓷鉴频器、高压发生器、红外探测器、声表面波器件、电光器件中的主要部件。</p><p><strong>2. 脉冲超声波的产生及其特点</strong></p><p>本实验中用于制造超声波换能器的压电陶瓷被加工成平面状,并在正反两面分别镀上银电极,称为压电品片。当给压电晶片两极施加一个电压短脉冲时,由于逆压电效应,晶片将发生弹性形变而产生弹性振荡,振荡频率与晶片的材料、厚度有关。可以通过改变晶片的厚度调整发生声波的频率,适当的晶片厚度可以得到超声频率范围的声波。在一次电压短脉冲作用下,晶片的振动幅度先迅速增大又迅速降低,因此,压电晶片发射出的是一个超声波波包,即通常所述的脉冲波,如图所示。</p><p><img alt="" height="280" src="/files/testpaper/106/2022/04-27/142014e507da086846.jpg" width="600" /></p><p>超声波在材料内部传播时,与被检对象相互作用发生散射,散射波被同一压电换能器接收,由于正压电效应,振荡的晶片在两极产生振荡的电压信号,该信号被放大后可以通过示波器检测。</p><p>t<sub>0</sub>是电脉冲施加在压电晶片的时刻,t<sub>1</sub>是超声波传播到试块底面又反射回来被同一个探头接收的时刻。如果不考虑探头延迟时间等其他因素的影响,超声波传</p><p>播到试块底面的时间为:t = ( t<sub>1&nbsp;</sub>- t<sub>0 </sub>)/2</p><p>如果试块材料均匀超声波波速 v 一定时,则超声波在试块中的传播距离为&nbsp; s=v * t</p><p><strong>3.超声波波型及其换能器种类</strong></p><p>如果压电晶片内部质点的振动方向垂直于压电晶片平面,那么换能器将向外发射的超声波为纵波。超声波在介质中传播通常有三种不同类型,并取决于介质可以承受何种作用力以及如何对介质激发超声波。</p><p>1) 纵波波型: 当介质中质点振动方向与超声波的传播方向一致时为纵波波型。任何固体介质当其体积发生交替变化时产生纵波。</p><p>2) 横波波型:&nbsp;当介质中质点的振动方向与超声波的传播方向相垂直时为横波波型。由于固体介质除了能承受体积形变外,还能承受切变形变,因此,当其有剪切力交替作用于固体介质时产生横波。需要指出的是横波只能在固体介质中传播。</p><p>3) 表面波波型:&nbsp;表面波是沿着固体表面传播的既具有纵波性质,又具有横波性质的超声波。最表面波可以看成是由平行于表面的纵波和垂直于表面的横波合成而来,在距表面1/4波长深处振幅强,随着深度的增加迅速衰减,在距离表面一个波长以上的地方,质点振动的振幅就十分微弱了。</p><p>在实际应用中,通常把超声波换能器称为超声波探头。常用的超声波探头有直探头和斜探头两种,其结构如图所示。</p><p><img alt="" height="260" src="/files/testpaper/106/2022/04-27/14205689d71d839035.jpg" width="600" /></p><p>探头通过保护膜或斜楔向外发射超声波,吸收背衬可以吸收晶片向背面发射的声波减少杂波的产生,匹配电感可以调整脉冲波的形状。</p><p>一般情况下,采用直探头产生纵波,斜探头产生横波或表面波。对于斜探头,晶片受刺激产生超声波后,声波首先在探头内部传播一段时间,才到达试块表面,这段时间称为探头的延迟。对于直探头,一般延迟较小,在测量精度要求不高的情况下,可以忽略不计。</p><p>在斜探头中,从晶片产生的超声波为纵波,它通过斜楔使超声波折射到试块内部,同时可以使纵波转换为横波。实际上,超声波在两种固体界面上发生折射和反射时,纵波可以折射和反射为横波,横波可以折射和发射为纵波,超声波的这种现象称为波型转换。</p><p><img alt="" src="/files/testpaper/106/2019/04-17/101959f1fbbc845197.jpg" /></p><p>式中,<span class="mq-math-mode" latex-data="\alpha\beta"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><var mathquill-command-id="3">&alpha;<sub>L</sub></var></span></span>和&nbsp;<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&alpha;</span><sub><span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-style: italic; white-space: nowrap; font-size: 15.3333px;">S&nbsp;</span></sub>分别是纵波反射角和横波反射角;<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&beta;<sub>L</sub></span>和&nbsp;<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&beta;</span><span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 15.3333px; font-style: italic; white-space: nowrap;"><sub>S</sub>&nbsp;</span><span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;"><sub>&nbsp;</sub></span>分别是纵波折射角和横波折射角;v<sub>1L&nbsp;</sub>和&nbsp;v<sub>1S</sub>分别是第一种介质的纵波声速和横波声速;v<sub>2L&nbsp;</sub>和&nbsp;v<sub>2S</sub>分别是第二种介质的纵波声速和横波声速。</p><p><strong>4. 超声波在固体中的传播</strong></p><p>在气体介质中声波只能是纵波,而在固体介质中由于固体可以承受切变形变,所以声波可以按纵波或横波两种波形传播。</p><p>在各向同性的固体材料中,应力和应变满足的胡克定律。</p><p>对于同一种固体材料(弹性介质),其纵波声速和横波声速的大小一般不同,而与材料自身的密度、弹性模量和泊松系数等弹性参数有关。固体在外力作用下,其长度沿力的方向产生变形,变形时的应力与应变之比就定义为弹性模量(E);在应力作用下,沿纵向有一正应变(伸长),沿横向就将有一个负应变(缩短),横向应变与纵向应变之比被定义为泊松系数(<span class="mq-math-mode" latex-data="\sigma"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><var mathquill-command-id="5">&sigma;</var></span></span><span>&nbsp;</span>)。</p><p style="margin: 0px;"><span lang="EN-US" style="margin: 0px; top: 6.5pt; font-family: 等线; font-size: 10.5pt; position: relative;"><v:shape id="_x0000_i1025" style="width:48.5pt;height:20.5pt" type="#_x0000_t75"></v:shape></span><img alt="" src="/files/testpaper/106/2019/04-17/005526e068f2378583.jpg" /><span lang="EN-US" style="margin: 0px; top: 6.5pt; font-family: 等线; font-size: 10.5pt; position: relative;"><v:shape style="width:48.5pt;height:20.5pt" type="#_x0000_t75">&nbsp;<v:imagedata chromakey="white" o:title="" src="file:///C:/Users/林雨嘉/AppData/Local/Temp/msohtmlclip1/01/clip_image007.png"></v:imagedata></v:shape></span></p><p>v<sub>L</sub>为材料中纵波声速;v<sub>S</sub>为材料中横波声速;E为弹性模量;<span class="mq-math-mode" latex-data="\sigma" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 13.075px;"><var mathquill-command-id="5">&sigma;</var></span></span>为泊松系数;<span class="mq-math-mode" latex-data="\rho"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><var mathquill-command-id="7">&rho;</var></span></span><span>&nbsp;</span>为材料的密度。</p><p>T为纵横波速度比,即T= v<sub>L&nbsp;</sub>/ v<sub>S</sub>。</p>

实验步骤

<p><strong>1. 观察仪器接收到超声回波(直探头与倾斜探头)</strong></p><p>1) 在铝试块CSK- IB表面加上适量的耦合剂(水)使用直探头的探测面与试块紧密接触。</p><p>2) 调节示波器,使其工作在YT耦合模式下,并调节扫描速率为10us/格 (可根据实际测量略做调整)、Y轴灵敏度为1V/格。</p><p>3) 观察直探头接收到的超声在试块中传播后的回波信号,并适当调节超声波实验仪上的衰减值,使得到的信号未达到饱和且有较明显的回波信号。</p><p>4) 更换斜探头,同样适当调节超声波实验仪上的衰减值,观察斜探头接收到的超声在试块中传播后的回波信号。</p><p><strong>2. 测量斜探头所产生的超声波(横波)入射铝块时的折射角</strong></p><p>使用铝试块CSK-IB上的A、B横通孔测量斜探头所产生的超声波(横波)入射铝块时的折射角,如图所示。</p><p><img alt="" height="450" src="/files/testpaper/106/2022/04-27/142116ce0877434198.jpg" width="600" /></p><p>分别使斜探头对准A孔和B孔,对准时示波器得到的信号为最大值,分别用钢板尺测量图中的L<sub>A1</sub>、L<sub>B1</sub>、H和L即可得到该超声横波的折射角<span class="mq-math-mode" latex-data="\beta"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><var mathquill-command-id="3">&beta;</var></span></span><sub>s</sub>。需要指出的是,铝试块CSK - IB的边界处也会反射超声,实验中应注意鉴别。</p><p><span class="mq-math-mode" latex-data="\beta" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 13.175px;"><var mathquill-command-id="3">&beta;</var></span></span><sub>s</sub>=arctan (( L<sub>B1</sub>-L<sub>A1</sub>-L ) / H )</p><p><strong>3. 铝块的弹性模量和泊松系数的测量</strong></p><p>1) 使用铝试块CSK - IB的厚度方向(其厚度为45mm)的一次回波测量直探头的延迟时间与纵波声波在试块中传播速度。其中 B<sub>1</sub> 是试块的一次底面回波,对应声波从发出传播到试块的下表面并反射回到接收端的信号;&nbsp;B<sub>2</sub> 是试块的二次底面回波,对应发出传播到试块的下表面并反射回到样品的上表面再次反射到下表面再反射回上表面并在接收器接收到的信号。分别从示波器上读出超声波(纵波)在试块内一次往复传播的时间 t<sub>1&nbsp;</sub>和两次往复传播时间 t<sub>2</sub>,则直探头的延迟时间为 t=2t<sub>1</sub>-t<sub>2&nbsp;</sub>。</p><p>在试块内纵波的声速为 v<sub>L</sub>=2L&#39; / ( t<sub>2</sub>-t<sub>1&nbsp;</sub>)。式中 L&#39;&nbsp; 为试块上下表面的距离。</p><p>2) 使用铝试块CSK -IB 的圆弧面测量斜探头的延迟时间与横波声波在试块中的传播速度,如图所示。</p><p><img alt="" height="450" src="/files/testpaper/106/2022/04-27/1424055b76de177142.jpg" width="600" /></p><p>使斜探头对准圆弧面,并使探头的斜射声波可以同时入射到 R<sub>1&nbsp;</sub>和 R<sub>2&nbsp;</sub>弧面上并在示波器上可以分别观察到其反射信号 B<sub>1&nbsp;</sub>和 B<sub>2</sub>,测量它们所对应的时间 t<sub>1</sub>和t<sub>2</sub>。</p><p>则斜探头的延迟时间为 t<sub>斜</sub>=(R<sub>2</sub>t<sub>1</sub>-R<sub>1</sub>t<sub>1</sub>)/(R<sub>2</sub>-R<sub>1</sub>)。</p><p>如果R<sub>2</sub>=2R<sub>1</sub>,则延迟时间为t<sub>斜</sub>=2t<sub>1</sub>-t<sub>2</sub>。</p><p>在试块内横波的声速为&nbsp; v<sub>S</sub>=2(R<sub>2</sub>-R<sub>1</sub>)/(t<sub>2</sub>-t<sub>1</sub>)。</p><p>将实验数据填写在表中,并用公式计算铝块的弹性模量和泊松系数。</p><p><strong>4.&nbsp; 超声成像</strong></p><p>1) 使用无缺陷超声成像试块(小),测量5M&Phi;6 直探头产生的超声在该试块材料中的传播速度。</p><p>2) 用 5M&Phi;6 专用直探头在有缺陷超声成像试块(大)上测量其缺陷分布,横纵分别每隔 10mm 测量一个实验点,分别记录探头的测量位置坐标以及缺陷波的接收时间。如果测量过程中,前面的回波明显高于噪声信号,则说明探头所探测位置内部存在缺陷。记录缺陷波的接收时间,并利用测得超声在该试块材料中的传播速度计算缺陷深度,将缺陷深度数据填写在表中。</p><p>3) 使用 Excel 的 &rdquo;曲面图&ldquo; 绘图功能或其他绘图工具绘制超声波成像专用试块的缺陷分布图。</p>

7.6 低真空的获得、测量与直流溅射法制备金属薄膜

实验目的和仪器

<p><strong>实验目的:</strong></p><p>(1)学习真空基本知识和真空的获得与测量技术基础知识;</p><p>(2)学习用直流溅射法制备薄膜的原理和方法;</p><p>(3)实际操作一套真空镀膜装置,使用真空泵和真空测量装置,研究该真空系统的抽气特性;</p><p>(4)用直流溅射法制备一系列不同厚度的金属薄膜,为实验研究金属薄膜厚度对其电阻率影响制备样品。</p><p><strong>实验仪器:</strong></p><p>SBC-12小型直流溅射仪(配有银靶)、机械泵、氩气瓶、超声波清洗器、玻璃衬底(长20mm、宽15mm、厚1mm)等。</p>

实验原理

<p><strong>1.真空基本知识及真空的获得与测量</strong><br />
(1)真空基本知识在真空技术里,真空是指相对大气而言,压强小于一个标准大气压的稀薄气体空间,常用&ldquo;真空度&rdquo;这个习惯用语和&ldquo;压强&rdquo;这一物理量来表示某一空间的真空程度。真空度越高,气体压强越低。<br />
通常气体的真空度直接用气体的压强来表示, 真空度的单位也采用压强的单位,常用帕斯卡(Pa)或托(Torr)作为单位。它们之间的换算关系为&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 1托(Torr)=1毫米汞柱(mmHg)=133帕斯卡(Pa)<br />
按气体压强大小的不同,通常把真空范围划分为:低真空1x10&deg;~1x102Pa,中真空1x10~1x10Pa,高真空1x10--1x10-Pa,超高真空1x10-5~1x10-Pa、极高真空1x10&deg;Pa以下。<br />
(2)空的获得空的获得就是人们常说的抽&ldquo;真空&rdquo;,是利用各种真空泵将被抽容器里的气体抽出,使该容器中的压强低于一个大气压。任何真空泵都不可能在整个真空范围内工作,通常真空泵按工作条件的不同可分为两大类:①可从一个大气压开始抽气的泵--&ldquo;前级泵&rdquo;,如机械泵、吸附泵。它们的极限真空度都不高(一般为1~10-2Pa)可实现低真空的获得或作为高一级真空泵的前级泵;②只能从较低的气压(通常是在I~10Pa真空度以下)开始抽气到更低压力的泵--&ldquo;次级泵&rdquo;,如扩散泵、分子泵、离子泵等。真空泵有两个重要参量:极限真空(真空泵能抽得的最高真空度)和抽气速率实际应用时要根据真空泵的有效使用范围,合理地选择真空泵。<br />
下面主要介绍机械泵和涡轮分子泵的结构与工作原理。<br />
1)机械泵:凡是利用机械运动(转动或滑动)来获得真空的泵,称为机械泵。机械泵有不同种类,其中以旋片式机械泵最为常见。图7.61是旋片式机械泵结构示意图。泵体主要由定子、转子、旋片、进气管和排气管等组成。定子两端被密封形成一个密封的泵体。泵腔内偏心地装有转子,它与泵内腔顶部紧密相切。沿转子的轴线开一个槽,槽内装有两块旋片,旋片中间用弹簧相连,弹簧使转子旋转时旋片始终沿定子内壁滑动。<br />
图7.6-2是旋片式机械泵工作原理图。旋片将泵腔分成了两部分。当转子转动时,A区体积缩小,压强增高,当压强高于一个大气压时,气体推开排气阀门排出气体;B区体积扩大,同时吸人气体(抽气);当B区到达最大位置时完成抽气过程,C 区开始抽气:当A区排气完毕,B区压缩排气,C区继续抽气。转子的不断转动使抽气、排气过程循环进行。机械泵是一种从一个大气压开始工作的真空泵,&quot;可以单独使用来获得低真空,也可以作为高真空泵或超高真空泵的前级泵。单级旋片泵的相极限真空可以达到1Pa,双级旋片泵的极限真空可以达到10<sup>-2</sup>Pa数量级。</p><p><strong>2.用直流溅射法制备薄膜的原理</strong><br />
&ldquo;溅射&rdquo;是指具有足够高能量的粒子(荷能粒子)轰击固体(称为靶材)表面,使固体表面的原子(或分子)从表面射出的现象。这些从固体表面射出的粒子大多呈原子状态,通常称为被溅射原子。常用的轰击靶材的荷能粒子为惰性气体离子(如氩离子)和其快速中性粒子,它们又被称为溅射粒子。溅射粒子轰击靶材,使靶材表面的原子离开靶材表面成为被溅射原子,这些被溅射出来的原子带有一定的动能,会沿着一定的方向射向衬底,沉积到衬底上就形成了薄膜,所以这种制备薄膜的方法被称为溅射法。我射法又可以细分为直流溅射法、磁控溅射法、射频溅射法和反应溅射法。<br />
溅射法基于荷能粒子轰击靶材时的溅射效应,而溅射过程都是建立在辉光放电的基础之上的,即溅射粒子都来源于气体放电。干燥气体在正常状态下是不导电的,是良好的绝缘介质,但当气体中存在自由带电粒子时,它就变为电的导体。这时若在气体中安置两个电极并加上电压,气体在强电场作用下,少量初始带电粒子与气体原子(或分子)相互碰撞,当碰撞能量超过某一临界值时,会使束缚电子脱离气体原子而成为自由电子。逸出电子后的原子成为正离子,使气体中的带电粒子增殖,这时有电流通过气体,这个现象称为气体放电。<br />
气体放电的根本原因在于气体中发生了电离的过程,在气体中产生了带电粒子。气体电离的基本形式有:<br />
(1)碰撞电离在电场作用下,那些散在气体中的带电粒子(电子或离子)被加速而获得动能,当它们的动能积累到一定数值后,在和中性的气体分子发生碰撞时,有可能使中性的气体分子发生电离,这种电离过程称为碰撞电离。在碰撞电离中,由于电子的尺小、重量轻,其平均自由行程也较大,所以在电场中容易被加速并积累起电离所需的值量因此,电子是碰撞电离中最活跃的因素,它在强电场中产生的这种碰撞电离是气体放电中带电粒子极为重要的来源<br />
(2)光电离由光辐射引起的气体分子的电离称为光电离。光子的能量与光的波长有关,波长越短,能量越大,各种短波长的高能辐射线如宇宙射线、&gamma;射线、X射线以及波紫外线等都具有较强的电离能力。<br />
(3)热电离因气体热状态引起的电离过程称为热电离。所有的气体都能发出热辐射,这也是电磁辐射。在高温下,热辐射光子的能量达到一定数值即可造成气体的热电离<br />
从基本方面来说,碰撞电离、热电离及光电离是一致的,都是能量超过某一临界值的粒子或光子碰撞分子使之发生电离,只是能量来源不同。在实际的气体放电过程中,这种电离形式往往同时存在,并相互作用。比如,在电场作用下,总会有碰撞电离发生。在放电过程中,当处于较高能位的激发态原子回到正常状态,以及异号带电粒子复合成中性位子时,又都会以光子的形式放出多余的能量,由此可能导致光电离,同时产生热能而引发热电离,高温下的热运动则又加剧碰撞电离过程。<br />
(4)表面电离气体中的电子也可以由电场作用下的金属表面发射出来,称为金属电极表面电离。从金属电极表面发射电子同样需要一定的能量,称为逸出功,它比气体的电离能小得多,所以金属电极表面发射电子要比直接使气体分子电离容易。可以用各种不同的方式向金属电极供给能量,如对阴极加热、正离子对阴极碰撞、短波光照射以及强电场等都可以使阴极发射电子<br />
气体放电的形式和现象是多种多样的,依气体压力、施加电压、电极形状、电源频率的不同,气体放电的形式总体上可以分为以下几类:①当气压较低,电源容量较小时,气原间的放电则表现为充满整个间隙的辉光放电;②在大气压下或者更高气压下,放电表现动跳跃性的火化,称为火花放电;③当电源容量较大且内阻较小时,放电电流较大,并出现高温的电弧,称为电弧放电;④在极不均匀电场中,还会在间隙击穿之前,只在局部电场很强的地方出现放电,但这时整个间隙并未发生击穿,这种放电称为局部放电,高压车电线路导线周围出现的电晕放电就属于局部放电;⑤当发生气体放电时,电极间交换的频车很高的放电形式叫高频放电;6此外,在气体放电中还有一种特殊的放电形式,即在介质与固体介质的交界面上沿着固体介质的表面而发生在气体介质中的放电,称为沿面版电,当沿面放电发展到使整个极间发生沿面击穿时称为沿面闪络,例如,在输电线路出现雷电过电压时,常常会引起沿绝缘子的表面的闪络<br />
辉光放电发生在较低气压气体中,如1-100Pa的低气压气体。不同的溅射技术所用国新光放电方式有所不同。直流溅射法利用的是直流电压产生的辉光放电;射频溅射法和用的是射频电磁场产生的辉光放电;磁控溅射法是利用平行于靶材表面的磁场控制下的电消电融场产生的辉光放电;反应溅射法可以利用惰性气体和活性气体的混合气体在电场电蛋场中产生的辉光放电,常用的活性气体有氧气、氮气等。<br />
本实验所使用的薄膜制备方法是直流溅射法,利用了直流电压产生的辉光放电。图6是直流溅射沉移装置示意图,其中溅射靶为阴极,衬底为阳极,阴极相对于阳极用领的数千伏电压。在对系统预抽真空以后,充人适当压力的惰性气体,例如10<sup>-1</sup>~10Pa的氩气作为气体放电的载体。在正、负极之间间的高压作用下,极间的气体分子(原子)被大量电离,并伴随发出辉光。图7.6-7 给出了靶材(阴极C)和衬底(阳极A)之间电位的变化情况,可以看到,两极间的电位的变化不是均匀的,有所谓的阴极电压降<br />
(Uc),即两极间的电压降主要发生在阴极附近。 由放电形成的气体正离子被朝着阴极&nbsp;<br />
(靶材)方向加速,并且由于两极间的电压降主要出现在靶材(阴极)附近,这些正离子和由其产生的快速中性粒子以它们在阴极电压降区域获得的几乎一样的能量(速度)到达靶材。阴极电压降的大小取决于气体的种类和阴极材料。在这些能量离子和中性粒子的轰击下,靶材原子被从其表面溅射出来,被溅射出来的靶材原子冷凝在阳极(衬底)上,从而形成了薄膜。直流溅射法适用于金属靶材。</p><p><img alt="" height="370" src="/files/testpaper/106/2022/04-21/235019b475eb502400.jpg" width="800" /></p><p><strong>3.直流溅射沉积装置</strong><br />
金属银具有较强的化学惰性,在空气中不氧化。采用银靶,可以在较低真空度下进行直流溅射而获得银金属薄膜,只需用机械真空泵提供1~2Pa的真空度即可,这样真空系统就大大简化了。本实验采用 SBC-12小型直流溅射仪,外观如图7.6-8所示。整个系统由真空系统和直流溅射镀膜系统组成。真空系统由一个直联旋片机械真空泵(2L/ s)和一个带石英观察窗的金属圆筒真空室组成,真空室内部结构如图<br />
7.6-9所示,金属圆筒与基座和顶盖间用橡胶圈密封。系统真空度由皮拉尼真空规和真空度显示仪表给出。溅射电压为2480V。镀膜时间用定时器控制,范围为10~110s。</p>

实验步骤

<p><strong>1.小型直流溅射镀膜仪真空系统真空特性测量</strong><br />
以小型直流溅射镀膜仪作为一个实际真空系统,通过实际操作,了解获得低真空的手段,熟悉抽真空的基本操作规程,观察和测量系统真空度随抽气时间的变化规律以及系统的极限真空。<br />
1)安装好小型直流溅射仪真空系统,轻轻左右转动金属圆筒和顶盖几次,使金属圆筒与基座和顶盖之间的橡胶密封圈密切接触,关闭顶盖上的放气阀和仪器面板上的针阀。<br />
2)打开小型直流溅射仪的&ldquo;总电源开关&rdquo;,机械泵开始对镀膜室抽真空,从真空表1定性观察镀膜室的真空度(气压)随抽气时间的变化趋势。<br />
3)根据定性观察趋势,制定定量测量镀膜室真空度随时间变化的数据测量点,记录镀膜室的真空度随抽气时间的变化,参见数据表格7.6-1,直到镀膜室真空度基本不变化为止。</p><p><strong>2、制备不同厚度的银薄膜</strong><br />
1)银靶装在镀膜室顶盖上。把银靶到工作台的距离调至40mm。将氩气瓶阀门打开。<br />
2)将玻璃衬底放人盛有无水乙醇的烧杯中,再将烧杯放人超声波清洗器中清洗3-5min。取出玻璃衬底,使玻璃衬底倾斜,以便无水乙醇流动时带走污物,并用吹风机彻底烘干玻璃衬底,然后放在镀膜室工作台中心位置。盖上镀膜室顶盖,轻轻左右转动金属圆筒和顶盖几次,使金属圆筒与基座和顶盖之间的橡胶密封圈密切接触。注意:彻底烘干玻璃衬底非常重要,否则镀出的银膜会氧化!<br />
3)打开&ldquo;电源&rdquo;开关,机械泵开始对镀膜室抽真空,从真空表上观测镀膜室的真空度。当真空度上升至20Pa时,溅射单元的&ldquo;准备&rdquo;灯亮,当真空度到达极限真空(不同镀膜装置略有差异,例如2~4Pa)时,打开氩气充气阀&ldquo;针阀&rdquo;(逆时针转动&ldquo;针阀&rdquo;为开,氩气流量加大;顺时针转动&ldquo;针阀&rdquo;为关,氩气流量减少。注意:调节&ldquo;针阀&rdquo;时,氩气流量变化会有所滞后,故调节&ldquo;针阀&rdquo;要缓慢。)向镀膜室中充入氩气,使镀膜室的气压较充气前增加1~2Pa。</p><p>4)设定好&ldquo;定时器&rdquo;的时间,即薄膜制备时间(溅射时间或沉积时间)。为防止过热,单次连续溅射的时间不要超过60s,制备较厚的薄膜时,可采取多次溅射的办法5)按下&ldquo;试验&rdquo;按钮,观察溅射电流大小,通过调节&ldquo;针阀&rdquo;使&ldquo;溅射电流表&rdquo;中显示的电流为5mA,立即松开&ldquo;试验&rdquo;按钮;然后,按下&ldquo;启动&rdquo;按钮,银靶被加上2480V的溅射电压,银薄膜沉积开始。这时,可以看见镀膜室内发出蓝色的辉光。&ldquo;溅射电流表&rdquo;显示的电流应稳定在5mA,可微调&ldquo;针阀&rdquo;控制氙气流量,保持溅射电流稳定。微调时要特别注意&ldquo;溅射电流表&rdquo;中显示的电流还能超过8mA,以免烧坏设备<br />
6)当&ldquo;定时器&rdquo;所设定的沉积时间达到之后,溅射电压降为零,溅射自动停止。&ldquo;溅射电流表&rdquo;显示的电流为零,完成了薄膜制备过程中的一次溅射沉积。如果需要多次溅射沉积,继续重复5)中所述内容,直到达到所需要的沉积时间,最终完成薄膜的制备。<br />
7)由于工作台温度较高(约70℃左右),需等待几分钟(待工作台温度降低再打开真空室,可避免银薄膜氧化),然后关闭氩气控制阀&ldquo;针阀&rdquo;,关上&ldquo;电源&rdquo;开关,开启镀膜室顶盖上的&ldquo;放气阀&rdquo;,给镀膜室放入空气(这时发出&ldquo;嗞嗞&rdquo;的声响)。镀膜室回到大气压下后(&ldquo;嗞嗞&rdquo;的声响消失),打开镀膜室上盖,取出薄膜样品。<br />
8)通过上述2)~7)的操作,制备出沉积时间为10min的银薄膜样品(在制备过程中,其他溅射条件均应保持一致),并将样品制备条件记录到表7.6-2中。<br />
注意:①不要使&ldquo;溅射电流表&rdquo;中显示的电流超过8mA,以免烧坏设备;②不要接触镀膜室顶盖上的电压输入端头,以免触电;③针阀关闭后应再反向拧两圈,以防止针阀长时间不用出现针阀回弹不灵活的现象。如遇到打开针阀时不向工作室放气,请将针阀石旋到底后再重新打开即可。</p>

5.17 太阳能电池

实验目的和仪器

<p><strong>【实验目的】</strong></p><p>(1)了解太阳能电池的光伏效应原理,了解单晶硅、多晶硅和非晶硅太阳能电池的差别;</p><p>(2)研究在无光照情况下太阳能电池的伏安特性(即暗伏安特性);</p><p>(3)研究在光照情况下太阳能电池的输出特性。</p><p><strong>【实验仪器】</strong></p><p>ZKY-SAC-I 太阳能电池特性实验仪、可变负载、光源、导轨、遮光罩、光强探头、单晶硅太阳能电池、多 晶硅太阳能电池、非晶硅太阳能电池。</p>

实验原理

<p><strong>1.太阳能电池光生伏特效应的原理&nbsp;</strong></p><p>&nbsp; &nbsp; 光生伏特效应是指半导体材料由于受到光照而产生电动势的现象,简称光伏效应。太阳能电池就是利用这 种半导体 P-N 结受到光照时的光伏效应进行发电的,其基本结构(如图 5.17-1 所示)就是一个大面积的平面 PN 结。</p><p>&nbsp; &nbsp; 当 P 型半导体与 N 型半导体结合形成 PN 结时,P 型半导体中的多子空穴和 N 型半导体中的多子电子在结 区相互扩散形成空间电荷区,并在空间电荷区内产生内建电场。电场会使载流子向扩散的反方向作漂移运动, 最终扩散与漂移达到平衡,使流过 P-N 结的净电流为零。在空间电荷区内,P 区的空穴被来自 N 区的电子复合, N 区的电子被来自 P 区的空穴复合,使该区内几乎没有能导电的载流子,因此空间电荷区又称为结区或耗尽区。 当入射光照射在 PN 结上时,由于内光电效应而产生光生载流子,及电子-空穴对。电子-空穴对在内建电场 的作用下分离,电子在内建电场的作用下进入 N 区,空穴在内建电场作用下进入 P 区,使 N 区有过量的电子而 带负电,P 区有过量的空穴而带正电,即在 PN 结两端形成了电压,如果将该 PN 结接入外电路,则该 PN 结就 可向负载输出电能。需要注意的是,太阳能电池产生光生伏特效应用于发电需要满足两个条件:(1)材料对光 具有本征吸收(可以产生内光电效应);(2)在太阳能电池内部可以形成内建电场,能够迅速分离光生载流子, 且能够阻止光生载流子的复合。&nbsp;</p><p><strong>2.太阳能电池的特性</strong></p><p>&nbsp; &nbsp; 当无光照射在太阳能电池时,可以将太阳能电池等效为一个二极管;有光照射在太阳能电池时,则可以将 其等效为一个受控电流源,其等效电路如图 5.17-2 所示。&nbsp;</p><p><img alt="" height="334" src="/files/testpaper/106/2022/04-24/192518e88db4437767.jpg" width="500" /></p><p>&nbsp; &nbsp; 图中,IL 为光照射到电池吸收层中产生的光生电流,当光照相对比较恒定的时候,光生电流不会随着工作 状态改变,可以看做恒流源。光生电压反向加在 PN 结两端,这样就会产生于光生电流反向的二极管反向饱和 电流光 ID。RL为负载电阻,V 为其电阻两端的电压,I 为电路中的电流。RP 为电池的并联电阻,主要是由于 PN 结漏电引起的,其中包括绕过电池边缘的漏电及由于结区存在晶体缺陷和外来杂质的沉淀物所引起的内部漏电。 RS 为电池的串联电阻,主要是由于电池材料所固有的电阻,所用电极材料的电阻,以及电池和电极相连部分由 于接触产生的电阻产生。 理想的太阳能电池正向电流 IF与其压降 UF之间满足以下关系式:</p><p>&nbsp;<img alt="" src="/files/testpaper/106/2022/04-24/19123756ac3b676532.jpg" /></p><p>&nbsp; &nbsp; 其中,I<sub>S </sub>为 PN 结的反向饱和电流,q 为电子电荷,k 为玻尔兹曼常数,T 为热力学温度。有无光照情况下太阳 能电池的伏安特性曲线,如图 5.17-3 所示,其中无光照时的伏安特性称为暗特性,有光照时称为光照特性。&nbsp;</p><p>&nbsp; &nbsp; 如果将太阳能电池的正负两级短路,其压降 UF变为 0,此时流过电路的电流称为太阳能电池的短路电流, 用 I<sub>SC </sub>表示。在理想条件下,太阳能电池的短路电流 I<sub>SC</sub> 与光电流 I<sub>L</sub>相等,即</p><p><img alt="" src="/files/testpaper/106/2022/04-24/19140194ff92983446.jpg" /></p><p>&nbsp; &nbsp; 其中,q 为电子电量,A 为 PN 结的横截面积,G 为由于外部作用(光照)所引起的净产生率,L<sub>e</sub>、L<sub>h</sub> 分 别为电子在材料中的扩散长度和空穴在材料中的扩散长度,W 为耗尽区宽度。 如果将太阳能电池的正负两级断开,其流过电流为 0,则此时太阳能电池两极见的压降 UF称为太阳能电池 的开路电压,用 UOC 表示。在理想条件下,太阳能电池的开路电压值为</p><p><img alt="" src="/files/testpaper/106/2022/04-24/191451bbea14528842.jpg" /></p><p><img alt="" height="447" src="/files/testpaper/106/2022/04-24/1926168af8de660367.jpg" width="500" />&nbsp; &nbsp;&nbsp;</p><p>&nbsp; &nbsp; 从图 5.17-3 中可以清晰的看到,当有光照射在太阳能电池上时,其输出功率 P 存在一个最大值 P<sub>max</sub>,此时 输出电压为 U<sub>mp</sub>,输出电流为 I<sub>mp</sub>。定义填充因子 F.F 为</p><p><img alt="" src="/files/testpaper/106/2022/04-24/191542e4de71345900.jpg" /></p><p>&nbsp; &nbsp; 填充因子是表征太阳电池性能优劣的重要参数,其值越大,电池的光电转换效率越高,一般的硅光电池 FF 值 在 0.75~0.8 之间。太阳能电池的转换效率可以定义为太阳能电池产生的能量和入射光的能量之比,即</p><p><img alt="" src="/files/testpaper/106/2022/04-24/1916248087b6924462.jpg" /></p><p>&nbsp; &nbsp;其中,P<sub>in</sub>为入射到太阳能电池表面的光功率。太阳能电池的转换效除了与其制造工艺有关外,还与照射在电池 上的光谱的性质,入射光的能量密度和太阳能电池的工作温度有关。对于太阳能电池来说,短路电流、开路电 压、填充因子以及转换效率是衡量其性能的主要参数。</p>

实验步骤

<p><strong>1. 太阳能电池的暗伏安特性测量</strong></p><p>&nbsp;太阳能电池的基本结构是一个大面积平面 P-N 结,单个太阳能电池单元的 P-N 结面积已远大于普通的二极 管。在实际应用中,为得到所需的输出电流,通常将若干电池单元并联。为得到所需输出电压,通常将若干已 并联的电池组串连。因此,它的伏安特性虽类似于普通二极管,但取决于太阳能电池的材料,结构及组成组件 时的串并连关系。太阳能电池的暗伏安特性是指无光照射时,流经太阳能电池的电流与外加电压之间的关系。 本实验提供的太阳能组件是将若干单元并联,测试时用遮光罩罩住太阳能电池。测试原理图如图 5.17-6 所 示。将待测的太阳能电池接到测试仪上的&ldquo;电压输出&rdquo;接口,电阻箱调至 50&Omega;后串连进电路起保护作用,用电 压表测量太阳能电池两端电压,电流表测量回路中的电流。&nbsp;</p><p>将电压源调到 0V,然后逐渐增大输出电压,每间隔 0.3V 记一次电流值,并将数据记录到表 5.17-1 中。将 电压输入调到 0V,并将&ldquo;电压输出&rdquo;接口的两根连线互换,即给太阳能电池加上反向的电压。逐渐增大反向电 压,每间隔 1V 记录一次电流值,并将数据记录到表 5.17-1 中。绘制三种太阳能电池的伏安特性曲线。</p><p><strong>&nbsp;2. 开路电压、短路电流与光强关系测量&nbsp;</strong></p><p>打开光源开关,并预热 5 分钟。取掉太阳能电池的遮光罩,将光强探头装在太阳能电池板位置,探头输出 线连接到太阳能电池特性测试仪的&ldquo;光强输入&rdquo;接口上。测试仪设置为&ldquo;光强测量&rdquo;。 由近及远移动滑动支架, 测量距光源一定距离的光强 Ilight,将测量到的光强记入表 5.17-2 中。</p><p>将光强探头换成单晶硅太阳能电池,测试仪设置为&ldquo;电压表&rdquo;状态。按图 5.17-7a 接线,按测量光强时的距 离值(光强已知),记录开路电压值于表 5.17-2 中。按图 5.17-7b 接线,记录短路电流值于表 5.17-2 中。重复该 实验步骤,分别将单晶硅、多晶硅以及非晶硅的开路电压与短路电流值记录在数据表格 5.17-2 中。绘制三种太 阳能电池的开路电压和短路电流随入射光强的变化曲线。&nbsp;</p><p><strong>3. 太阳能电池的输出特性测量&nbsp;</strong></p><p>按图 5.17-8 接线,以电阻箱作为太阳能电池负载。在一定光照强度下(将滑动支架固定在导轨上某一个位 置),分别将三种太阳能电池板安装到支架上,通过改变电阻箱的电阻值,记录太阳能电池的输出电压 U、电流&nbsp;I 以及当前的负载值,将测量数据填于表 5.17-3 中(需测量 10 组以上的实验点),并计算太阳能电池的输出功 率 P<sub>O</sub>=U&times;I。&nbsp;</p><p>根据表 5.17-3 数据作 3 种太阳能电池的输出伏安特性曲线及功率曲线,找出最大功率点,对应的电阻值即 为最佳匹配负载。</p><p>&nbsp;分别由式 5.17-4 和式 5.17-5 计算三种太阳能电池的填充因子和转换效率。其中,入射到太阳能电池板上的 光功率 P<sub>in</sub>=I<sub>light</sub>&times;S<sub>1</sub>,I<sub>light</sub> 为入射到太阳能电池板表面的光强,S<sub>1</sub>为太阳能电池板面积(约为 50mm&times;50mm)。&nbsp;</p><p><img alt="" src="/files/testpaper/106/2022/04-29/1025393a7b2a585165.jpg" /></p>

补充实验1——金属丝线胀系数的测定

实验目的和仪器

<div><p><strong>【实验目的】</strong></p><p>测量金属管的长度随温度的变化曲线,并得到该金属的线膨胀系数。&nbsp;</p><p><strong>【实验仪器】</strong></p><p>钢卷尺、游标卡尺、温度计、控温炉、光杠杆、望远镜、金属管等。</p>

实验原理

<p>金属膨胀现象:当金属的温度升高时,由于内部分子热运动加剧,其分子间距离加大,导致使宏观线度增加。</p><p>设某金属管在 0℃时,长度 L<sub>0 </sub>;&nbsp;当温度升至 t<sub>1&nbsp;</sub>时,其长度为 L<sub>1</sub>,则 𝐿<sub>1 </sub>= 𝐿<sub>0&nbsp;</sub>(1 + 𝛼𝑡<sub>1</sub>)&nbsp;</p><p>式中,<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&alpha;&nbsp;</span>为金属管的线膨胀系数。在温度变化不太大的情况下,<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&alpha;&nbsp;</span>视为常数,大小只与金属的性质有关。</p><p>当温度升至&nbsp; t<sub>2&nbsp;</sub>时,金属管的长度为 𝐿<sub>2</sub> = 𝐿<sub>0&nbsp;</sub>(1 + 𝛼𝑡<sub>2</sub>)&nbsp; 故可得&nbsp;𝛼 =( 𝐿<sub>2</sub>&nbsp;&minus; 𝐿<sub>1</sub>)&nbsp;/ ( 𝐿<span style="font-size: 13.3333px;"><sub>0</sub>&nbsp;(&nbsp;</span>𝑡<sub>2</sub>&nbsp;&minus; 𝑡<sub>1</sub>))</p><p>因线膨胀系数 𝛼 很小(其数量级为 10<sup>-5 </sup>/ ℃),故当 t<sub>1</sub>为室温时,可得出 L<sub>1&nbsp;</sub>&asymp; L<sub>0</sub></p><p>有&nbsp;𝛼 = ( 𝐿<sub>2</sub>&nbsp;&minus; 𝐿<sub>1</sub>)&nbsp;/ ( 𝐿<span style="font-size: 13.3333px;"><sub>1</sub>&nbsp;(&nbsp;</span>𝑡<sub>2</sub>&nbsp;&minus; 𝑡<sub>1</sub>))= ∆𝐿 / (𝐿<sub>1&nbsp;</sub>( 𝑡<sub>2</sub> &minus; 𝑡<sub>1</sub>))</p><p>金属管的微小伸长&nbsp;∆𝐿 是由光杠杆来测量的。由光杠杆原理可知:∆𝐿 = 𝑏 (𝑠<sub>2</sub>&nbsp;&minus; 𝑠<sub>1</sub>) / ( 2&nbsp;𝐷 )</p><p>式中 b 为光杠镜后脚尖到两前脚尖的垂直距离,D 为光杠镜面到标尺的垂直距离,𝑠<sub>1</sub>、𝑠<sub>2&nbsp;</sub>分别为当金属管的温度为 𝑡<sub>1&nbsp;</sub>及 𝑡<sub>2&nbsp;</sub>时望远镜的读数。</p><p>故最终可得: 𝛼 = &nbsp;𝑏 (𝑠<sub>2</sub>&nbsp;&minus; 𝑠<sub>1</sub>) / ( 2&nbsp;𝐷 ( 𝐿<sub>1&nbsp;</sub>( 𝑡<sub>2</sub>&nbsp;&minus; 𝑡<sub>1&nbsp;</sub>)))&nbsp;</p><p>实验装置如图所示:</p><p><img alt="" height="245" src="/files/testpaper/106/2022/05-20/193955b99cae945532.jpg" width="400" /></p>

实验步骤

<p><strong>1. 用钢卷尺测量金属管长度。</strong></p><p><strong>2. 用游标卡尺测量光杠杆臂长。&nbsp;</strong></p><p><strong>3. 将金属管放置到加热装置中。&nbsp;</strong></p><p><strong>4. 将温度计放置在金属管上。&nbsp;</strong></p><p><strong>5. 将光杠杆放到仪器的平台上,调整光杠杆的镜面铅直。&nbsp;</strong></p><p><strong>6. 用钢卷尺测量光杠杆镜面到标尺的距离。&nbsp;</strong></p><p><strong>7. 调节尺读望远镜&nbsp; &nbsp; &nbsp;</strong></p><p>&nbsp; &nbsp; &nbsp;(1)&nbsp;调节高度调节旋钮,使望远镜与光杠镜等高;&nbsp;</p><p>&nbsp; &nbsp; &nbsp;(2) 调节目镜调节旋钮,使视野中叉丝至最清晰状态;&nbsp;</p><p>&nbsp; &nbsp; &nbsp;(3)&nbsp;调节调焦旋钮,使视野中的像至最清晰状态;</p><p>&nbsp; &nbsp; &nbsp;(4)&nbsp;调整望远镜的位置,使视野中出现标尺。</p><p><strong>8. 打开加热装置,调整功率旋钮,加热金属管,测量从室温到 100℃之间待测金属的线膨胀系数。设定多个温度点,将第 1 次温度达到平衡时的温度及标尺读数分别作为 T<sub>1</sub>,s<sub>1</sub>。提高温度,在新值达到平衡后,记录当前温度 T 及标尺读数于表中。</strong></p><p><strong>9. 关闭加热装置,使金属管降温。</strong></p><p><strong>10.&nbsp; 提交数据,整理仪器,结束实验测量。</strong></p>

补充实验2——偏振光的观察与研究

实验目的和仪器

<p><strong>【实验目的】</strong></p><p>(1)了解光的偏振现象;&nbsp;</p><p>(2)了解偏振片、波晶片的作用;&nbsp;</p><p>(3)了解光的检偏方法。</p><p><strong>【实验仪器】</strong></p><p>&nbsp; &nbsp;所光源、偏振片(2 个)、&lambda;/4 波晶片、&lambda;/2 波晶片、光功率计等。</p>

实验原理

<p><strong>1. 光的偏振状态&nbsp;</strong></p><p>光有五种偏振状态,即线偏振光、椭圆偏振光、圆偏振光、自然光和部分偏振光。</p><p>平面偏振光或线偏振光:在传播过程中,电矢量的振动方向始终在某一确定方向。</p><p>自然光:光源发射的光对外不显现偏振的性质。</p><p>部分偏振光:在发光过程中,光的振动面在某个特定方向上出现的几率大于其他方向,即在较长时间内电矢量在某一方向上较强。&nbsp;</p><p>椭圆偏振光或圆偏振光:振动面的取向和电矢量的大小随时间作有规律的变化,电矢量末端在垂直于传播方向的平面上的轨迹是椭圆或圆。</p><p><img alt="" height="207" src="/files/testpaper/106/2022/05-26/1109106f420d263306.jpg" width="400" /></p><p>其中线偏振光和圆偏振光可看作椭圆偏振光的特例,而椭圆偏振光可看作是两个沿同一方向 z 传播的振动方向相互垂直的线偏振光的合成。</p><p><img alt="" height="374" src="/files/testpaper/106/2022/05-26/11095530e7e0186233.jpg" width="400" /></p><p>偏振光的振幅 A 可以分解为 x 方向上的分量 E<sub>x&nbsp;</sub>和 y 方向的分量 E<sub>y</sub>,则 E<sub>x</sub> 和 E<sub>y</sub>分别可由以下两式表示:</p><p>E<sub>x&nbsp;</sub>=A<sub>x</sub> cos( <span class="mq-math-mode" latex-data="\omega"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><var mathquill-command-id="3">&omega;t - kz</var></span></span>)</p><p>E<sub>y&nbsp;</sub>=A<sub>y</sub> cos(&nbsp;&nbsp;<span class="mq-math-mode" latex-data="\omega" style="font-size: 18.4px;"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1" style="width: 52.7625px;"><var mathquill-command-id="3">&omega;t - kz&nbsp;</var></span></span>+<span class="mq-math-mode" latex-data="\varepsilon"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><var mathquill-command-id="5">&epsilon;</var></span></span> )&nbsp;</p><p>式中 A 为振幅,<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&omega;&nbsp;</span>为两光波的圆频率,t 表示时间,k 为波矢量的数值,<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&epsilon;&nbsp;</span>为两波的相对位相差。</p><p><strong>2. 波晶片(旋光晶体)</strong></p><p>利用单轴晶体的双折射,所产生的寻常光(o 光)和非常光(e 光)都是线偏振光。当一束单色平行自然光正入射到波晶片上,光在晶体内部便分解为 o 光与 e 光。o 光在晶体内的波速为 V<sub>o</sub> , e 光为 V<sub>e</sub>,即相应的折射率 n<sub>o</sub>、n<sub>e</sub>不同。</p><p>设晶片的厚度为 L, 则两束光通过晶片后有位相差:</p><p><span class="mq-math-mode" latex-data="\delta=\frac{2\pi}{\ \lambda\ }"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><var mathquill-command-id="3">&delta;</var><span class="mq-binary-operator" mathquill-command-id="5">=</span><span class="mq-fraction mq-non-leaf" mathquill-command-id="9"><span class="mq-numerator" mathquill-block-id="10"><span mathquill-command-id="6">2</span><span class="mq-nonSymbola" mathquill-command-id="7">&pi;</span></span><span class="mq-denominator" mathquill-block-id="11"><span mathquill-command-id="20">&nbsp;</span><span class="mq-nonSymbola" mathquill-command-id="12">&lambda;</span><span mathquill-command-id="19">&nbsp;</span></span><span style="display:inline-block;width:0">​</span></span></span></span>( n<sub>o</sub>-n<sub>e&nbsp;</sub>)L</p><p>其中&nbsp;<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 16.56px; text-align: center; white-space: nowrap;">&lambda;</span> 为光波在真空中的波长。<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&delta;</span>= 2kp 的晶片,称为全波片;<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&delta;</span>=2kp &plusmn;&nbsp;<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 16.56px; text-align: center; white-space: nowrap;">&pi;</span> 时,为半波片;<span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&delta;</span> = 2k&pi; &plusmn; <span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 16.56px; text-align: center; white-space: nowrap;">&pi;</span>&nbsp;/2 为四分之一波片。</p><p><strong>3. 偏振光的检测&nbsp;</strong></p><p>按照马吕斯定律,强度为 I<sub>0</sub> 的线偏振光通过检偏器后,透射光的强度为<span class="mq-math-mode" latex-data="\cos2\Theta"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span></span></p><p>I = I<sub>0</sub><span class="mq-math-mode" latex-data="\cos\left(\Theta\right)2"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span></span><span class="mq-math-mode" latex-data="\cos^2\Theta"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><var class="mq-operator-name" mathquill-command-id="87">c</var><var class="mq-operator-name" mathquill-command-id="88">o</var><var class="mq-operator-name" mathquill-command-id="89">s</var><span class="mq-supsub mq-non-leaf mq-sup-only" mathquill-command-id="83"><span class="mq-sup" mathquill-block-id="84"><span mathquill-command-id="86">2</span></span></span><span mathquill-command-id="90">&Theta;</span></span></span><span>&nbsp;</span></p><p>式中,<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Theta;&nbsp;</span>为入射光偏振方向与检偏器的偏振轴之间的夹角。</p><p>当以光线传播方向为轴转动检偏器时,透射光强度 I 将发生周期性变化。当 <span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Theta;</span>= 0 时,透射光强度为极大值;当 <span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Theta;</span>= 90<sup>o</sup> 时,透射光强度为极小值,即消光状态,接近于全暗;当 0&lt;&nbsp;<span style="font-family: Symbola, &quot;Times New Roman&quot;, serif; font-size: 18.4px; white-space: nowrap;">&Theta;&nbsp;</span>&lt;&nbsp;90<sup>o</sup>&nbsp;时,透射光强度 I 介于最大值和最小值之间。故根据透射光强度变化的情况,可以区别线偏振光、自然光和部分偏振光。</p><p>自然光通过起偏器和检偏器的变化如图所示:</p><p><img alt="" height="234" src="/files/testpaper/106/2022/05-26/111014637292661813.jpg" width="400" /></p>

实验步骤

<p>研究四分之一波片对偏振光的影响&nbsp;</p><p>(1)使偏振片 A 和 B 的偏振轴正交(消光)。然后插入&nbsp;<span class="mq-math-mode" latex-data="\lambda"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><span class="mq-nonSymbola" mathquill-command-id="3">&lambda;</span></span></span><span>&nbsp;</span>/ 4 波片 C(实际实验中要使光线尽量穿过元件的中心)。&nbsp;</p><p>(2)以光线为轴先转动 C 使消光,使正交偏振片处于消光状态时&nbsp; 1/4&nbsp; 波片的光轴位置作为 0&deg; 位置(注意:此 0&deg;位置并非指 1/4 玻片上的 0 刻度),然后使 B 转过 360&deg;,在表格 2 中记录下转过的角度与对应的光强值。</p><p>(3)再将 C 从消光位置转过 30&deg;、45&deg;,重复步骤 (2), 分别在表格 2 中记录下转过的角度与对应的光强值,并绘制曲线。</p>

补充实验3——拉曼光谱实验

实验目的和仪器

<p><strong>【实验目的】</strong></p><p>(1)了解拉曼现象的产生原理;&nbsp;</p><p>(2)学会用拉曼光谱学的基本测试技术。&nbsp;</p><p><strong>【实验仪器】</strong></p><p>拉曼光谱仪,激光,CCl<sub>4</sub> 样品、C<sub>6</sub>H<sub>6</sub> 样品等</p>

实验原理

<p><strong>1. 拉曼现象</strong></p><p>当光照射到物质上时会发生非弹性散射,散射光中除有与激发光波长相同的弹性成分(瑞利散射)外,还有比激发光波长长的和短的成分,后一现象统称为拉曼效应。由分子振动、固体中的光学声子等元激发与激发光相互作用产生的非弹性散射称为拉曼散射,一般把瑞利散射和拉曼散射合起来所形成的光谱称为拉曼光谱。</p><p>设散射物分子原来处于基电子态,振动能级如图所示。</p><p><img alt="" height="264" src="/files/testpaper/106/2022/06-02/08413934cf80191744.jpg" width="400" /></p><p>散射光中既有与入射光频率相同的谱线,也有与入射光频率不同的谱线,前者称为瑞利线,后者称为拉曼线。在拉曼线中,又把频率小于入射光频率的谱线称为斯托克斯线,而把频率大于入射光频率的谱线称为反斯托克斯线。&nbsp;</p><p>拉曼光谱的示意图如图所示。</p><p><img alt="" height="254" src="/files/testpaper/106/2022/06-02/084315341d04994168.jpg" width="400" /></p><p>一般将瑞利线与拉曼线的波数差称为拉曼频移,即散射光(即测量到的谱线)相对于原激发光光源频率的差值。</p><p><br />
<span class="mq-math-mode" latex-data="\Delta"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><span mathquill-command-id="3">&nbsp; &Delta;v =&nbsp;</span></span></span><span class="mq-math-mode" latex-data="\omega"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><var mathquill-command-id="5">&omega;<sub><span style="font-size: 15.3333px;">入</span></sub></var></span></span><sub style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-style: italic; white-space: nowrap; font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">射光&nbsp;</sub><span class="mq-math-mode" latex-data="\omega"><span class="mq-root-block" mathquill-block-id="1">-&nbsp;</span></span><span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 18.4px; font-style: italic; white-space: nowrap;">&omega;<sub>散</sub></span><sub style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-style: italic; white-space: nowrap; font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">射光</sub><span class="mq-math-mode" latex-data="\omega"><span class="mq-root-block" mathquill-block-id="1">&nbsp; =&nbsp;</span></span><span class="mq-math-mode" latex-data="\frac{107}{\lambda}"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><span class="mq-fraction mq-non-leaf" mathquill-command-id="7"><span class="mq-numerator" mathquill-block-id="8"><span mathquill-command-id="11">1</span><span mathquill-command-id="12">0</span><sup><span mathquill-command-id="13">7</span></sup></span><span class="mq-denominator" mathquill-block-id="9"><span class="mq-nonSymbola" mathquill-command-id="14">&lambda;<sub>入射光</sub>(nm)</span></span><span style="display:inline-block;width:0">​</span></span></span></span><span>&nbsp;</span><span class="mq-math-mode" latex-data="\omega"><span class="mq-root-block" mathquill-block-id="1">-</span></span><span class="mq-math-mode" latex-data="\frac{107}{\lambda}" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 125.912px;"><span class="mq-fraction mq-non-leaf" mathquill-command-id="7" style="font-size: 16.56px;"><span class="mq-numerator" mathquill-block-id="8"><span mathquill-command-id="11">1</span><span mathquill-command-id="12">0</span><sup><span mathquill-command-id="13">7</span></sup></span><span class="mq-denominator" mathquill-block-id="9" style="width: 115.312px;"><span class="mq-nonSymbola" mathquill-command-id="14">&lambda;<sub>散射光</sub>(nm)</span></span><span style="display: inline-block; width: 0px;">​</span></span></span></span>&nbsp;<span class="mq-math-mode" latex-data="\omega" style="font-size: 18.4px;"><span class="mq-root-block" mathquill-block-id="1" style="width: 27.5px;">&nbsp;</span></span></p><p>其中拉曼频移的单位用波数(cm<sup>-1</sup> )来表示。</p><p>需要指出的是,样品的拉曼频移不随激光光源频率的变化而发生变化,无论采用什么波长的激光照射样品,其产生的拉曼频移的多少是固定的。 可以说拉曼频移是分子振动能级的直接量度。&nbsp;</p><p><strong>2. 拉曼效应的经典电磁解释</strong></p><p>对于拉曼效应来说,当一束激发光( 现在通常使用激光 )照射在物体上时,光子与物体分子碰撞有可能发生能量交换,光子将一部分能量传递给分子或从分子获得一部分能量,从 而改变光波的频率( 光谱发生位移 )。由于散射光的频率是入射光频率 f<sub>0</sub> 与分子振动固有频率的联合,故拉曼散射又称联合散射。</p><p>设入射光电场为 E = E<sub>0&nbsp;</sub>cos( 2&pi;f<sub>0</sub>t ),分子因电场作用产生的偶极矩为</p><p>P = &epsilon;<sub>0&nbsp;</sub>&chi; E&nbsp;</p><p>其中,&chi; 为分子极化率。若 &chi; 为不随时间变化的常数,则 P 以入射光 f<sub>0</sub> 作周期性变化,由此得到的散射光频率也为 f<sub>0</sub>,这就是瑞利散射。若分子以固有频率 f 振动,此时分子极化率不再为常数,也随 f 做周期变化,可表示为 &chi;=&chi;<sub>0</sub>+&chi;<sub>(f) </sub>cos( 2&pi;ft )。式中 &chi;<sub>0</sub> 为分子静止时的极化率;&chi;<sub>(f)</sub> 为相应于分子振动所引起的变化极化率的振幅。可得:</p><p>P = &epsilon;<sub>0</sub>&chi;<sub>0</sub>E<sub>0&nbsp;</sub>cos( 2&pi;f<sub>0</sub>t )+&epsilon;<sub>0</sub>&chi;<sub>(f)</sub>E<sub>0&nbsp;</sub>cos( 2&pi;f<sub>0</sub>t ) cos( 2&pi;ft )&nbsp;</p><p>&nbsp; &nbsp;= &epsilon;<sub>0</sub>&chi;<sub>0</sub>E<sub>0&nbsp;</sub>cos( 2&pi;f<sub>0</sub>t ) + &epsilon;<sub>0</sub>&chi;<sub>(f)</sub>E<sub>0&nbsp;</sub>[cos2&pi;(f<sub>0</sub>+f)t + cos2&pi;(f<sub>0</sub>-f)t]</p><p>该式表明,感应电偶极距 P 的频率有三种 f<sub>0</sub>、f<sub>0</sub>-f&nbsp; 和&nbsp; f<sub>0</sub>+f,所以实验上测得的散射光的频率也有三种。频率为 f<sub>0 </sub>的谱线为瑞利散射线,频率为 f<sub>0</sub>-f 的散射线为斯托克斯线,频率为 f<sub>0</sub>+f 的散射线为反斯托克斯线。&nbsp;</p><p><strong>3. 当光入射到样品上时的三种情况</strong></p><p>(1) 光子同样品分子发生了弹性碰撞,没有能量交换,只是改变了光子的运动方向,此时散射光频率 = 入射光频率,即<span class="mq-math-mode" latex-data="hv_k"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><var mathquill-command-id="42">&nbsp;h</var><var mathquill-command-id="55">v</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="51"><span class="mq-sub" mathquill-block-id="52"><var mathquill-command-id="54">k</var></span><span style="display:inline-block;width:0">​</span></span></span></span><span>&nbsp;&nbsp;</span>=&nbsp;&nbsp;<span class="mq-math-mode" latex-data="hv_1"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><var mathquill-command-id="42">h</var><var mathquill-command-id="41">v</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="37"><span class="mq-sub" mathquill-block-id="38"><span mathquill-command-id="40">1</span></span><span style="display:inline-block;width:0">​</span></span></span></span><span>&nbsp;</span><span>,如图所示:</span></p><p><img alt="" height="426" src="/files/testpaper/106/2022/06-02/084339bb2f7b643666.jpg" width="400" /></p><p>(2) 如频率为&nbsp;&nbsp;<var mathquill-command-id="41" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif; white-space: nowrap;">v</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="37" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; font-family: Symbola, &quot;Times New Roman&quot;, serif; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="38" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><span mathquill-command-id="40" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">1&nbsp;&nbsp;</span></span></span>的入射光子被样品吸收,样品分子被激发到能量为&nbsp;<var mathquill-command-id="42" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif; white-space: nowrap;">&nbsp;h</var><var mathquill-command-id="55" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif; white-space: nowrap;">v</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="51" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; font-family: Symbola, &quot;Times New Roman&quot;, serif; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="52" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><var mathquill-command-id="54" style="font-size: inherit; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif;">L&nbsp;</var></span></span>&nbsp;的振动能级 <span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 16.56px; font-style: italic; white-space: nowrap;">L</span>=1 上,同时发生频率为&nbsp;<var mathquill-command-id="55" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif; white-space: nowrap;">v</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="51" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="52" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><font face="Times New Roman, Symbola, serif"><i>S</i></font><font face="Symbola, Times New Roman, serif">&nbsp;=&nbsp; </font></span></span><var mathquill-command-id="41" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif; white-space: nowrap;">&nbsp;v</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="37" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; font-family: Symbola, &quot;Times New Roman&quot;, serif; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="38" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><span mathquill-command-id="40" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">1</span></span></span><span class="mq-supsub mq-non-leaf" mathquill-command-id="51" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="52" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><font face="Symbola, Times New Roman, serif">&nbsp;</font></span></span><var mathquill-command-id="55" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif; white-space: nowrap;">- v</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="51" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; font-family: Symbola, &quot;Times New Roman&quot;, serif; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="52" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><var mathquill-command-id="54" style="font-size: inherit; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif;">L&nbsp;</var></span></span>的斯托克斯散射,如图所示 :&nbsp;&nbsp;</p><p><img alt="" height="404" src="/files/testpaper/106/2022/06-02/084410a9632c592219.jpg" width="400" /></p><p>(3) 如果分子处于振动能级为 <span style="font-family: &quot;Times New Roman&quot;, Symbola, serif; font-size: 16.56px; font-style: italic; white-space: nowrap;">L</span>=1 的激发态,入射光子吸收了这一振动能级的能量就会发生频率为&nbsp;<var mathquill-command-id="55" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif; white-space: nowrap;">v</var><sub><span class="mq-math-mode" latex-data="\alpha"><span class="mq-textarea"><textarea autocapitalize="off" autocomplete="off" autocorrect="off" spellcheck="false" x-palm-disable-ste-all="true"></textarea></span><span class="mq-root-block" mathquill-block-id="1"><var mathquill-command-id="56">&alpha;</var></span></span></sub><span class="mq-supsub mq-non-leaf" mathquill-command-id="51" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="52" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><font face="Times New Roman, Symbola, serif"><i>S</i></font><font face="Symbola, Times New Roman, serif"> =&nbsp;</font></span></span><var mathquill-command-id="41" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif; white-space: nowrap;">&nbsp;v</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="37" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; font-family: Symbola, &quot;Times New Roman&quot;, serif; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="38" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><span mathquill-command-id="40" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box;">1</span></span></span><span class="mq-supsub mq-non-leaf" mathquill-command-id="51" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="52" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><font face="Symbola, Times New Roman, serif">&nbsp;</font></span></span><var mathquill-command-id="55" style="font-size: 18.4px; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif; white-space: nowrap;">+ v</var><span class="mq-supsub mq-non-leaf" mathquill-command-id="51" style="font-size: 16.56px; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: inline-block; vertical-align: -0.5em; font-family: Symbola, &quot;Times New Roman&quot;, serif; white-space: nowrap;"><span class="mq-sub" mathquill-block-id="52" style="font-size: inherit; line-height: inherit; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; display: block; float: left;"><var mathquill-command-id="54" style="font-size: inherit; line-height: 0.9; margin: 0px; padding: 0px; border-color: black; user-select: none; box-sizing: border-box; font-family: &quot;Times New Roman&quot;, Symbola, serif;">L&nbsp;</var></span></span>的反斯托克斯散射,如图所示:</p><p><img alt="" height="474" src="/files/testpaper/106/2022/06-02/084429df04d0755567.jpg" width="400" /></p>

实验步骤

<p><strong>1、&nbsp;</strong>打开光谱仪、打开激光,调整光谱仪的外光路。</p><p><strong>注意三个要点:</strong></p><p>a.&nbsp; 观察到瑞利光的成象清晰,并进入摄谱仪的入射狭缝;</p><p>b. 调整聚光部件,使汇聚光的腰部正好位于样品管中心,从各个方面观察,激光束都应通过样品的中心;</p><p>c. 开始测试前需关闭箱盖。&nbsp;</p><p><strong>2、&nbsp;</strong>关闭激光光源,进行阈值分析,将设备阈值分析后的数值填入到软件界面中;</p><p><strong>3、&nbsp;</strong>在软件界面处选择扫描模式为 &rdquo;波数模式&ldquo;;</p><p><strong>4、&nbsp;</strong>打开光谱仪箱盖,选择样品为 CCl<sub>4</sub>,采集 CCl<sub>4</sub> 分子的振动拉曼光谱。打开激光光源,精细调节光谱仪的外光路,得到 5个 CCl<sub>4</sub> 斯托克斯散射峰,记录下 CCl<sub>4</sub> 的拉曼光谱曲线。</p>

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